# maximal_combinations_of_fairness_definitions__4c32bf60.pdf Journal of Artificial Intelligence Research 82 (2025) 1495-1579 Submitted 07/2024; published 03/2025 Maximal Combinations of Fairness Definitions Mary Beth Defrance marybeth.defrance@ugent.be Tijl De Bie tijl.debie@ugent.be IDLAB, Ghent University 9000 Ghent, Belgium The so-called Impossibility Theorem for fairness definitions is one of the more striking research results with both theoretical and practical consequences, as it states that satisfying certain combinations of fairness definitions is impossible. To date, this negative result has not yet been complemented with a positive one: a characterization of which combinations of fairness notions are possible. This work aims to fill this gap by identifying maximal sets of commonly used fairness definitions for binary classification that can be simultaneously satisfied. The fairness definitions used are demographic parity, equal opportunity, predictive equality, predictive parity, false omission rate parity, overall accuracy equality and treatment equality. We conclude that in total 12 maximal sets of these fairness definitions are possible, among which are seven combinations of two definitions, and five combinations of three definitions. Our findings also shed light on the practical relevance and utility of each of these 12 maximal fairness definitions in various scenarios, regarding the accuracy of the classifier and ratios of false positives and false negatives, considering the base rates. 1. Introduction The field of fairness in AI started gaining more attention after the Pro Publica article about the recidivism risk assessment system COMPAS (Angwin et al., 2016). Due to this article, questions arose on which characteristics an AI system needs to satisfy for it to be fair in a practical, intuitive, or legal sense. One of the main criticisms from the authors of the Pro Publica article was the difference between the false positive and false negative rates for black people compared to white people. The creators of the COMPAS-tool defended their system by arguing that it was properly calibrated, meaning that defendants with the same scores had similar rearrest rates for both demographic groups, and that this made the system fair. One could argue that ideally these properties would both be satisfied. However, the works of Chouldechova (2016), Barocas et al. (2019), and Kleinberg et al. (2016) all concluded that these properties cannot be satisfied simultaneously. This result is colloquially referred to as the Impossibility Theorem , stating it is mathematically impossible to achieve calibration, equal false positive and equal false negative rates simultaneously, except in a practically irrelevant, degenerate case. (See Sec. 2 for further details.) The usage of fairness definitions has often been limited to enforcing one at a time. This severely limits the complexity of the fairness requirements imposed on a system. Creating a contrasting Possibility Theorem might open up the possibility of simultaneously imposing multiple compatible fairness definitions. These combinations could allow to enforce multiple perspectives on fairness simultaneously (Rahman et al., 2024a; Park et al., 2022; Luo et al., 2024). Additionally, it would also indicate which perspectives are incompatible. 2025 The Authors. Published by AI Access Foundation under Creative Commons Attribution License CC BY 4.0. Defrance & De Bie Contributions To the best of our knowledge, this negative result has never been complemented with a positive one, namely the characterisation of maximal sets of fairness notions that can be simultaneously satisfied. In this paper, we investigate which and how many of the more frequently used fairness definitions can be combined, leading to a set of maximal notions of fairness on which no additional of the commonly used fairness notions can be imposed without making the problem infeasible. We do this for the simplest but most commonly studied case of binary classification. Out of a range of seven commonly used fairness definitions (Demographic Parity (Dwork et al., 2012), Equal Opportunity (Hardt et al., 2016), Predictive Equality (Hardt et al., 2016), Predictive Parity (Chouldechova, 2016), False Omission Rate Parity, Overall accuracy Equality (Berk et al., 2021) and Treatment Equality (Berk et al., 2021)), we identify a total of 12 maximal combinations, including seven maximal combinations of two and five maximal combinations of three fairness definitions. Figure 1 shows the possible combinations of fairness definitions. In addition, we investigate the constraints these combinations impose on the accuracies of both groups and on the confusion matrices. We find that imposing a combination of two or more fairness definitions imposes a strict behaviour on the relation of errors made between two sensitive groups. This behaviour is influenced by the base rates of both groups, which means that for some datasets a combination of fairness definitions would heavily constrain the performance for one sensitive group. 2. Related Work The field of fairness in AI knows two large, distinct research topics. The first one is based on the explainability or interpretability of AI systems. This research aims to create AI models that are explainable or to create explanations for existing models in order to investigate if the models exhibit unfairness, which is identifiable for people. This approach is related to the legal concept of procedural fairness (Australian Law Reform Commission, 2016), meaning that the decision process itself should be deemed fair. The second research focus, adopted in the current paper, concerns the formulation and study of mathematical definitions of fairness. The aim is to have the model adhere to a certain fairness notion, defined in terms of certain mathematical properties on the outcome of the model. This method if often referred to as outcome fairness and relates to the legal concept of distributive justice (Lamont & Favor, 2017), meaning that the distribution of the outcome decision should be deemed fair. Both approaches are not mutually exclusive, but work with a different ideology on how fairness should be achieved. It is possible to have a fair procedure, thus procedural fairness, that results in an unfair distribution and vice versa. In this section, we briefly elaborate on both these research lines. 2.1 Procedural Fairness Approaches: Explainability and Interpretability The simplest approach in explainability is to only use non-discriminatory features. For example, Grgic-Hlaca et al. (2018) let human participants decide which features are acceptable to base a prediction on given a specific context. A special case of this fair feature selection is fairness through unawareness, wherein no sensitive attributes are included in the features. Maximal Combinations of Fairness Definitions Figure 1: The possible sets of fairness definitions are represented by nodes, according to Definition 1. The nodes with an asterisk are maximal sets which cannot be extended with an extra fairness definition, whilst satisfying Definition 1. Defrance & De Bie A second explainability approach to AI fairness focuses on the output behaviour of a model, and aims to assess the fairness of outputs per individual. This can be achieved by interchanging the values of the sensitive attributes, and checking if doing so changes the prediction (Agarwal et al., 2018; Galhotra et al., 2017). Counterfactuals extend this principle by analysing the model s behaviour through the difference between the real-world data and a counterfactual one. This counterfactual data is created by changing the sensitive attribute and using a causal model to subsequently adjust the other features (Kusner et al., 2017). This approach is favoured to account for the correlation between the sensitive attribute and the other features. Other methods besides causal models can also be used to adjust for these correlations (Black et al., 2020). The work of Grari et al. (2023) extends the capabilities of counterfactuals to also function with continuous sensitive variables. Another method using counterfactuals is investigating the difference in effort certain groups would have to make in order to change their predictions (Sharma et al., 2020). The work of Goethals et al. (2023) has a similar goal: they investigate which attributes have an imbalanced influence on the prediction for a group with a certain sensitive attribute. 2.2 Outcome Fairness: Fairness Definitions on the Output Distribution The research around fairness definitions can be categorised in two main groups. The first group consists of methods enforcing a certain fairness definition on a learned model, often called fairness methods. The other consist of more conceptual research on these definitions. In general this group is about either the introduction of new fairness definitions, surveys on fairness definitions, the interpretation of previously proposed fairness notions, or about socalled impossibility theorems showing the incompatibility of certain combinations of fairness definitions. Here we survey the work most directly related to the present paper. 2.2.1 Surveys on Fairness Definitions Verma & Rubin (2018) discuss different fairness definitions for classification and focus on creating a human-understandable interpretation of these definitions and applying them to a specific use case. An elaborate discussion about the design of a predictive model concerning pitfalls and biases that occur in the process can be found in Mitchell et al. (2021). It also covers different fairness definitions and relates the choice between these definitions back to the discussion of design choices. They touch upon the impossibilities of combining certain fairness definitions when working with scores instead of binary decisions. The survey of Mehrabi et al. (2021) contains an extensive list of possible biases and fairness definitions. The fairness definitions are grouped by type, group, subgroup or individual. It also discusses methods to satisfy them. A similar recent survey was done by Caton & Haas (2024). Pessach & Shmueli (2022) published a similar survey. They conclude with several emerging research sub-fields of algorithmic fairness. Ruf & Detyniecki (2021) strive to create a type of Fairness Compass which should help practitioners decide what fairness definition is best suited to a given problem. Throughout the work, different fairness definitions are clearly explained through the use of confusion matrices and with a theoretical use case. Congruently, a similar tool was developed with the Aequitas Toolkit (Saleiro et al., 2019). However, their Fairness Tree is constructed based on the use of the system rather than the properties in the data, on which the Fairness Compass is designed. Maximal Combinations of Fairness Definitions 2.2.2 Prior Work on Combining Fairness Definitions The majority of research on combining fairness definitions focusses on the impossibility of certain combinations. Chouldechova (2016) discuss how adherence to a certain fairness criterion can lead to considerable disparate impact between groups. They state it is impossible to achieve equality between groups for false positive rate (FPR), positive predictive value (PPV) and false negative rate (FNR) when the base rates differ between groups. Kleinberg et al. (2016) discuss combining three specific fairness definitions in the context of risk scoring. The work focusses on scoring rather than binary classification. They conclude that combining those three definitions is only possible under unique circumstances, namely if the groups have equal base rates or if the model is capable of perfect prediction. Berk et al. (2021) focus on the trade-off that would occur when using multiple incompatible fairness definitions such as discussed in the work by Chouldechova (2016). They also provide different techniques for achieving this goal. Related to the two original Impossibility Theorems, Beigang (2023) created a new impossibility theorem, which concerns the combination of counterfactual fairness with equalized odds and predictive parity. They also provide some relaxations to the fairness definitions in order for them to be compatible. Bell et al. (2023) investigated the impossibility theorems in practice. In this work, experiments are used to investigate the effects of enforcing equal FPRs, FNRs and PPVs on the accuracy of the system. A similar result is found in the work by Hsu et al. (2022). They use integer programming to solve this multi-objective problem. In their experiment they also reaffirm the existing impossibility theorems. The work of Rosenblatt & Witter (2023) shows that counterfactual fairness and demographic parity are basically equivalent. Pleiss et al. (2017) created a relaxed version of equalised odds and calibration for which the combination is feasible in more situations. However, they indicate themselves that the models which satisfy this combination of constraints have limited usefulness. Another approach was taken by Rahman et al. (2024b). They designed a novel set of fairness definitions. The paper focusses on intra-marginal and intersectional concepts of fairness and express these concepts for both individual and group fairness. 2.2.3 Combining Fairness Definitions with Other Metrics The work of Cummings et al. (2019) shows that given a system with non-trivial accuracy, it is impossible to combine pure differential fairness with strict equal opportunity. Trivial accuracy is the maximal accuracy a system could achieve when only predicting the same label. They also show that combining differential fairness with a relaxed version of equal opportunity is feasible. A subsequent work of Agarwal (2021) refutes this possibility of the combination with a relaxed notion of fairness. Furthermore, they prove that pure differential fairness is impossible to combine with either demographic parity, equal opportunity or equalised odds while achieving non-trivial accuracy. The work of Pinz on et al. (2023) finds that given a probabilistic data source, equal opportunity is incompatible with non-trivial accuracy under certain conditions. This is complementary to the work of Hardt et al. (2016), which proves that equal opportunity and non-trivial accuracy are compatible for a deterministic data source. Defrance & De Bie 3. Maximal Combinations of Fairness Definitions Common fairness definitions can be expressed as functions of the variables in confusion matrices (such as Table 1) for the different protected demographic groups. More specifically, they are defined as the equality of a statistic, such as the false negative rate, computed on the confusion matrices for the different demographic groups. Often, these demographic groups are based on their sensitive attributes. We articulate these expressions for seven common fairness definitions in Sec. 3.1. This approach makes combining fairness definitions easy: simply combining the constraints they impose on the confusion matrices. Our analysis considers two groups, referred to as groups a and b. However, later we show that our findings hold up for multiple groups, if these groups are disjoint. Note that confusion matrices commonly use counts. However, we normalize the confusion matrices such that the four variables in each confusion matrix sum to 1, as this simplifies the calculations. Table 1: Symbolic representation of a confusion matrix. Predicted Positive Negative True Positive TP FN Negative FP TN The values in the confusion matrix are constrained due to properties of the dataset itself: TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y 0 TPa 1 x, 0 TPb 1 y 0 TNa x, 0 TNb y 0 FPa x, 0 FPb y 0 FNa 1 x, 0 FNb 1 y The variables 1 x and 1 y in Equation (1) denote the base rates of group a and group b respectively. The base rate is the fraction of positive samples in the data per group. The inequalities will be left out of subsequent derivations to keep the notations concise. Let us now define what we mean when we say that a certain combination of fairness definitions is possible . Before doing this, the following two observations are relevant. First, if the base rates of the two groups are equal, i.e. if x = y, then it is always possible to satisfy any combination of fairness definitions, as the confusion matrices can then be identical. If the confusion matrices are identical, then any statistics computed on them will also be equal. However, base rates are a property of the dataset and hence situation dependent. Thus, any definition of what it means for a combination of fairness definitions to be possible , should be independent of the base rates. Second, some combinations of fairness definitions may be too constraining that jointly enforcing them leads to certain variables in the confusion tables having a fixed (and possibly Maximal Combinations of Fairness Definitions trivial) value. Although this means that the combination can strictly speaking be satisfied, it leaves so little freedom to the classifier that it becomes of limited practical use. Thus, we propose a definition where also such combinations are not deemed possible . The following definition determines the necessary properties for a combination of fairness definitions to be deemed possible . Definition 1 (Possibility of combining fairness definitions). A combination of fairness definitions is deemed possible if after combining the constraints, all elements in both confusion matrices can still take on values within a subset of [0, 1] with non-zero Lebesgue measure, and this for all pairs of base rates (1 x, 1 y) from a subset of [0, 1]2 with non-zero 2-dimensional Lebesgue measure. Or informally speaking, the elements of the confusion matrices should be able to take on values within a non-trivial range, and this for all base rate pairs from within a non-trivial 2-dimensional range. Remark 1. This definition is solely focussed on finding some configuration of the confusion matrix that will satisfy the fairness definitions. It would also be possible to include the performance of the resulting classifier as a requirement. This was done by Barocas et al. (2019). They additionally required that the prediction of the model must not be independent of the ground truth. In other words, it would require the classifier not to be random. Remark 2. This paper only considers satisfying fairness definitions, meaning the equality must be satisfied. Other approaches to fairness aim to minimize the difference between two relevant quantities, instead of requiring an equality. In this line of research, several methods have been proposed to use this difference as a loss term in order to make a model more fair (Kamishima et al., 2011; Zemel et al., 2013; Padala & Gujar, 2020; Buyl et al., 2024) . The results in the present paper remain valid for this approach as an incompatibility will signify that minimizing to a difference of zero is impossible, thus signalling that the minimization task will be working on conflicting goals. 3.1 Fairness Definitions Considered in the Present Paper The present paper considers the following fairness definitions: demographic parity, equal opportunity, predictive equality, predictive parity, false omission rate parity, overall accuracy equality and treatment equality. Table 2 contains two equivalent definitions for each fairness definition: one expressed in terms of probabilities, and another in terms of a constraint on the confusion matrices. The table also notes the statistical property it requires to be equal between groups (if it has a standard name). In its last column, it states the orientation of the constraint on the confusion matrices (horizontal, vertical, or diagonal). Other frequently used fairness definitions exist, particularly in other settings than binary classification, but this paper focusses only on the binary prediction setting. A well known example is calibration as is used in Kleinberg et al. (2016) for risk scoring, its equivalent for binary classification, Predictive Parity, is used in this work. Note that each of these definitions only imposes one constraint on the confusion matrix, combining them increases the number of constraints. Below we discuss each of the considered fairness definitions in greater detail. Figure 2 illustrates them more visually, highlighting the orientations on the confusion matrix, and thus showing their differences and similarities. Defrance & De Bie Table 2: The list of fairness definitions considered in this paper, defined in probabilistic terms (second column) and defined in terms of elements in the confusion matrices (third column). The fourth column names the statistic that is constrained to be equal across the confusion matrices. The fifth column mentions the orientation of the constraints on the confusion matrix as shown in Figure 2. Probabilities Confusion Matrix Statistic Orientation Demographic Parity P( ˆY = y|A = 0) = P( ˆY = y|A = 1) FPA+TPA FPA+TPA+FNA+TNA = FPB+TPB FPB+TPB+FNB+TNB Positive Rate (PR) Board (Bv) Equal Opportunity P( ˆY = 1|A = 0, Y = 1) = P( ˆY = 1|A = 1, Y = 1) TPA TPA+FNA = TPB TPB+FNB True Positive Rate (TPR) Horizontal (H) Predictive Equality P( ˆY = 1|A = 0, Y = 0) = P( ˆY = 1|A = 1, Y = 0) FPA FPA+TNA = FPB FPB+TNB False Positive Rate (FPR) Horizontal (H) Predictive Parity P(Y = 1|A = 0, ˆY = 1) = P(Y = 1|A = 1, ˆY = 1) TPA TPA+FPA = TPB TPB+FPB Pos. Prediction Value (PPV) Vertical (V) False Omission Rate Parity P(Y = 1|A = 0, ˆY = 0) = P(Y = 1|A = 1, ˆY = 0) FNA FNA+TNA = FNB FNB+TNB False Omission Rate (FOR) Vertical (V) Overall accuracy Equality P( ˆY = Y |A = 0) = P( ˆY = Y |A = 1) TPA+TNA TPA+FPA+TNA+FNA = TPB+TNB TPB+FPB+TNB+FNB Accuracy (ACC) Board (Bd) Treatment Equality FNA FPB - Diagonal (D) Maximal Combinations of Fairness Definitions Definition 2 (Demographic Parity - DP (Dwork et al., 2012)). In order to satisfy demographic parity both groups proportionally must receive the positive prediction (TP + FP) equally regardless of the ground truth and thus also regardless of the base rates. This is unlike other fairness definitions, the result of which is that satisfying demographic parity can come at the cost of predictive performance. Conditional statistical parity (Corbett-Davies et al., 2017) is a similar fairness definition, which requires equal positive rates between sensitive groups when restricted to people that share a certain attribute value. Conditional statistical parity will not be considered in this paper. Definition 3 (Equal Opportunity - EOP (Hardt et al., 2016)). Equal opportunity is both a definition of itself and also one of two conditions that is required for the popular fairness definition equalised odds (Hardt et al., 2016). Equal opportunity requires that people have the same probability of receiving a negative prediction given that they belong in the positive category, regardless of their sensitive attribute. Definition 4 (Predictive Equality - PE (Hardt et al., 2016)). Predictive equality is the second condition required in equalised odds, alongside equal opportunity. It requires that the probability of receiving a positive prediction when the ground truth is negative is equal across demographic groups. If both predictive equality and equal opportunity are satisfied then equalised odds is satisfied. Definition 5 (Predictive Parity - PP (Chouldechova, 2016)). Predictive parity is yet another kind of definition as it is conditional on the probability of the actual class and not of the predicted class. Expressed on the confusion matrix, it imposes a vertical ratio on it rather than a horizontal one like equal opportunity and predictive equality. In order to satisfy predictive parity the probability that a positive prediction is correct must be equal across sensitive groups. Definition 6 (False Omission Rate Parity - FORP). False omission rate parity is similar to predictive parity. It requires that the probability of a negative prediction to be correct is equal across sensitive groups. The combination of satisfying both predictive parity and false omission rate parity is also called conditional use accuracy equality (Verma & Rubin, 2018). Definition 7 (Overall accuracy Equality - Oa E (Berk et al., 2021)). Overall accuracy equality requires equal classifier accuracy between sensitive groups. Although straightforward and intuitive, this definition alone is often insufficient to guarantee intuitive fairness. Indeed, it implicitly assumes an equal importance of positive and negative predictions, which is often inaccurate in practice. Definition 8 (Treatment Equality - Tr. Eq. (Berk et al., 2021)). Treatment equality is an atypical fairness definition as it cannot be expressed as a probability nor as a commonly used statistic. It is a constraint on the false positives and false negatives, requiring their ratio to be equal for sensitive groups. Defrance & De Bie Treatment Equality(--) Equal Opportunity (-) Demographic Overal accuracy Parity (..) False Omission Rate Parity (..) Predictive Equality (-) Figure 2: Orientations of the fairness definitions in the confusion matrix. 3.2 Combining Two Fairness Definitions We begin with the pairwise combinations of the fairness definition discussed in Sec. 3.1. Proposition 1. All pairwise combinations of demographic parity, equal opportunity, predictive equality, predictive parity, false omission rate parity, overall accuracy equality and treatment equality are possible in accordance with Definition 1. Proof outline. To prove for a given pair of fairness definitions that they can be combined pairwise, we pair the constraints of the confusion matrices, as listed in Eq. (1), and the constraints from the respective fairness definitions to yield a combined system of equations. We solve this system of equations to express a subset of the variables in terms of the remaining free variables. The base rates must be such free variables, as required in Def. 1. An example of calculating the constraints for combining two fairness definitions is shown in Eq. (2), where demographic parity is combined with equal opportunity: TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa TPa+FNa = TPb TPb+FNb TPa+FPa 1 = TPb+FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa 1 x = TPb 1 y TPa+FPa 1 = TPb+FPb FNa = 1 x 1 x 1 y TPb TNa = x x y 1 y TPb FPb FNb = 1 y TPb TNb = y FPb TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb (2) By combining this result with the inequalities from Eq. (1), feasible ranges for each of the free variables and the base rates can be computed. For conciseness, a shortened version of the calculation of these ranges is shown in Eq. (3), while the full system of inequalities can be found in Appendix A.1.1: 1 y TPb 1 x 1 x 1 y TPb 1 x 1 y TPb FPb x x y 1 y TPb FPb x 1 y TPb + FPb x y 1 y TPb + FPb x 1 1 y TPb 1 1 x 1 y TPb 0 FPb x + x+y 1 y TPb x+y 1 y TPb FPb x+y 1 y TPb FPb FPb x + x+y Maximal Combinations of Fairness Definitions (2) DP & Oa E DP & Tr. Eq. (3) EOP & PP(3) (3) PE & FORP (3) PE & Oa E (7) PP & FORP (6) PP & Oa E FORP & Oa E Oa E & Tr. Eq. EOP & Tr. Eq. PP & Tr. Eq. PE & Tr. Eq. FORP & Tr. Eq. FP = TP + FP 1 - y x - y 1 - y a b b TP = FP + TP FP = FP a b x FP = FP + a b 2 TP = TP + a b FN = FN ( +1) a b FP - FN b b FP = FP ( +1) a b FP - FN b b 1 - x 1 - y a b FP = FP TP = TP 1 - y a b 1 - x FP = FP 1 - y a b 1 - x FN = FN 1 - y a b 1 - x TN = TN 1 - y a b 1 - x TP = TP 1 - y a b 1 - x TP = (TN - TN ) y - x b b 1 - y TN = TN x y a b FN = FN x y a b TN = (TP - TP ) x x - y a b a TN = (TP - TP ) y x - y b b a TN b FN . TN a TP = FP ( +1) b b y - x FP - FP a b TP = FP ( +1) a a y - x FP - FP a b FN = FN ( +1) a b y - x TN - FN b b TN = TN ( +1) a b x - y TN - FN b b FP = FP Figure 3: Resulting constraints when combining pairs of fairness definitions. This system of inequalities shows a specific range for the free variables FPb, TPb, and both base rates 1 x and 1 y are unconstrained. The range for FPb is [max(0, x+y 1 y ), min(x + x+y 1 y , y)] which simplifies to [0, x + x+y 1 y ] if x y, which means it has a non-zero Lebesgue measure, or [ x+y 1 y , y] if y > x then TPb will have a non-zero Lebesgue measure if x > y2. Thus the Lebesgue measure is non-zero for each free variable for a set of base rates with a non-zero 2-dimensional Lebesgue measure, meaning that the combination of demographic parity with equal opportunity is possible. The proofs for the other pairwise combinations of fairness definitions follow a similar structure and can be found in Appendix A. The results are summarized in Figure 3. From Proposition 1 we can infer that none of the fairness definitions are in contradiction with each other. In other words, what is fair according to one of the fairness definitions considered, is not impossibly fair according to another. Figure 3 shows that the combinations of fairness definitions result in a unique set of constraints. However, two exceptions exist: the three pairwise combinations of any two fairness definitions taken from the set (EOP, PP, Tr. Eq.) are identical to each other, and the same is true for all pairwise combinations of fairness definitions from the set (PE, FORP, Tr. Eq.). The interpretability of these combinations differs greatly. The solution of the combination of predictive parity with false omission rate parity is too tedious to detail in Table 3, but can be found in the calculations of combining predictive parity, false omission rate parity and overall accuracy equality in Section B.25. Defrance & De Bie Table 3: The structure of the seven different constraint types and the associated combination of orientations. Nr. Orientation of the combined definitions Constraint structure (1) Bv H Xa = Factor BR Xb Yb = Factor BR Xb + Yb (2) Bv V , Bd D Xa = Xb Ya = Yb (3) HH, HV, HD, V D Xa = Factor BR Xa Ya = Factor BR Yb (4) Bv D, V Bd Xa = Factor BR,Xb,Yb Xb Ya = Factor BR,Xb,Yb Yb (5) Bv Bd Xa = Xb + Factor BR Ya = Yb + Factor BR (6) V V Xa = Factor BR,Ya,Yb Yb+ Factor BR,Ya,Yb Ya Xb = Factor BR,Ya,Yb Yb+ Factor BR,Ya,Yb Ya (7) HBd Xa = Factor BR (Yb Ya) Xb = Factor BR (Yb Ya) We point out that some structure can be found in the resulting constraints from the pairwise combinations. Seven types of constraint combinations can be discerned, as denoted in the top right corner of the combinations in Figure 3. Which of these seven types a pairwise combination belongs to, is dependent on the orientations of the fairness definitions as discussed in Section 3.1, Table 2 and Figure 2. A summary of the structures and the corresponding orientations of the fairness definitions can be found in Table 3. 3.2.1 The P%-Rule We additionally investigate a relaxed fairness definition based on demographic parity, namely the p%-rule (Roth et al., 2021). Instead of requiring equality of the positive rates, as in demographic parity, the p%-rule requires that positive rate in any demographic group cannot be less than p% of the positive rate in any other demographic group. Formally: a, b : PRgroup a PRgroup b p 100. (4) The p%-rule is unique as it is based on a legal concept, the 80%-rule, from the Equal employment opportunity commission (1978). It requires that any selection rate for recruitment procedures is at least 4/5 of the group with the highest selection rate. If this is not satisfied then it is seen as adverse impact. In other words it requires that the positive rates between groups must satisfy the p%-rule, in this instance p = 80. There are two special instances of the p%-rule, namely if p = 0 or p = 100. If p = 0 then no restrictions are placed on the system as a ratio is always larger than 0. The second possibility of p = 100 requires for the positive rates to be equal across all groups, which is equivalent to requiring demographic parity. Because each fairness definition can be combined with demographic parity and the p%-rule with p < 100 is a relaxation of demographic parity, all fairness definitions can be combined with the p%-rule. Maximal Combinations of Fairness Definitions 3.2.2 Extending to Multiple Demographic Groups Thus far only two demographic groups were taken into consideration. However, the analysis can easily be extended to multiple disjoint groups. When combining any two fairness definitions, two degrees of freedom remain in the system of linear equations. It is then possible to parameterize these equations in such a way that those two free variables represent two of the four variables in the confusion matrix of just one demographic group. Let us refer to this demographic group as the first one. The variables in the confusion matrix of the other demographic group are then expressed through those two free variables. Then adding an additional demographic group can be done by simply constraining its confusion matrix in the same manner. This ensures that the resulting fairness notions are equal to those from the first demographic group. As the fairness notions in each additional demographic group are equal to those of the first group, they will be equal to each other. In order for this to hold, it is necessary that the demographic groups are disjoint to prevent dependencies between groups. Disjoint groups can be achieved by creating a new group for each possible combination of sensitive attributes. Depending on the number of sensitive attributes, this could result in a large number of groups and lead to some very small groups. The work of Ghosh et al. (2021) argues that this intersectional fairness may be preferable for detecting every type of unfairness. We therefore argue that this property of extending to multiple sensitive groups is sufficient for most contexts. 3.3 Combining Three Fairness Definitions We characterize the set of possible combinations of three fairness definitions. Proposition 2. When combining three fairness definitions out of demographic parity, equal opportunity, predictive equality, predictive parity, false omission rate parity, overall accuracy equality, and treatment equality only five out of 35 possible combinations are feasible (according to Definition 1). Proof outline. Section 3.2 shows that combinations of three fairness definitions are possible as certain pairwise combinations are equivalent. To examine other possible combinations of three fairness definitions, we combine the constraints of two pairwise combinations which share one fairness definition. Equation (5) shows an example of the calculations for combining demographic parity, equal opportunity, and predictive equality: TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb TPa = y x y FPb + TPb FPa = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x 1 y TPb FPb = y 1 y TPb TPb = TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb x = y 0 = 0 Defrance & De Bie The resulting constraints show that one variable, TPb, is free, the base rates are unconstrained, and none of the variables are fixed to a particular value. Therefore, the combination of demographic parity, equal opportunity and predictive equality is possible. Remark 3. Note that the prediction of any classifier that satisfies demographic parity, equal opportunity, and predictive equality, is independent of the ground truth: the probability of any combination of a ground truth label value and predicted label value is equal to the marginal probabilities of these values. Indeed, independence can be expressed in terms of the variables in the confusion matrix: TPa = (TPa + FPa) (TPa + FNa), TPb = (TPb + FPb) (TPb + FNb), TNa = (TNa + FNa) (TNa + FPa), TNb = (TNb + FNb) (TNb + FPb). It is easy to verify that given Eq. (5), this set of equations expressing independence is satisfied. While this is of course undesirable behaviour of a classifier, Definition 1 does not deem such combinations impossible , because it is possible to satisfy the equations in non-trivial value ranges albeit only with an essentially random classifier. It is worth reiterating that e.g. in Barocas et al. (2019), such combinations were defined as incompatible as well. We return to this point in Sec. 3.6. However, not all combinations of three fairness definitions are possible. The combination of demographic parity, predictive parity, and false omission rate parity serves as an example. The calculations can be found in Eq. (7): TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb TNa = TNb FNa = FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb x FPa = y FPa x TPa = y TPa FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb x = y x = y (7) The final set of constraints requires x = y, which violates the condition of the base rates having a non-zero 2-dimensional Lebesgue measure. Therefore, these three fairness definitions can only be achieved simultaneously when the base rates are equal, which is entirely datadependent. The combination of demographic parity, predictive parity and false omission rate parity is thus not possible. Maximal Combinations of Fairness Definitions Combining demographic parity, equal opportunity and predictive parity is also impossible, as shown in Eq. (8): TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb TPa = TPb FPa = FPb FNa = 1 x TPa TNa = x FPa FNb = 1 y TPb TNb = y FPb x = y TPa = TPb TPa = TPb FPa = FPb FNa = 1 x TNa = x FPa FNb = 1 y TNb = y FPb TPb = 0 TPa = 0 TPa = 0 FPa = FPb Eq. 8 shows that the solution either requires equal base rates or it requires for TPb and TPa to equal zero. The occurrence of the former was previously discussed in this section, how this leads to incompatibility. In that latter case, the feasible set of four out of eight variables in the confusion matrices will have a Lesbesgue measure of zero, which violates the requirements for being a possible combination from Definition 1. The final kind of outcome, when combining three fairness definitions, arises when combining predictive parity, false omission rate parity and overall accuracy equality. In this case, the following second degree polynomial needs to be satisfied: ( 2x + 1)TN2 b + (xy y + x + 2y TNa x2 + 2x TNa 2TNa)TNb +(xy x + TNa 2y TNa + y y2)TNa = 0. (9) As this equation is not transparent, a numerical approach was used to solve it. The results are plotted in Figure 4. From Figure 4 we can see that a free variable, e.g. TNa, exists. It is solvable for a range of base rates, although not for all base rates. However, this range of base rates consists of non-trivial real intervals and therefore satisfies Definition 1. The plots show that none of the variables have a Lebesgue measure of zero, such that the combination of predictive parity, false omission rate parity and overall accuracy equality is possible. (a) Relation between TNa and TNb (b) Relation between TNa, TPa and TPb Figure 4: Relation between the free variable TNa and the variables TNb, TPa, TPb for (PP, FORP, Oa E), showing multiple values are possible for TPb and TPa given TNa. Defrance & De Bie Table 4: Summary of possible and impossible combinations of three fairness definitions. Possible combinations Impossible combinations (DP, EOP, PE) (DP, PP, FORP) (DP, EOP, PP) (DP, FORP, Tr. Eq.) (EOP, PE, Oa E) (DP, PP, Oa E) (DP, EOP, FORP) (DP, Oa E, Tr. Eq.) (EOP, PP, Tr. Eq.) (DP, FORP, Oa E) (DP, EOP, Oa E) (EOP, FORP, Tr. Eq.) (PE, FORP, Tr. Eq.) (DP, PP, Tr. Eq.) (DP, PE, Tr. Eq.) (EOP, PE, Tr. Eq.) (PP, FORP, Oa E) (DP, PE, PP) (EOP, PE, PP) (EOP, PE, FORP) (EOP, Oa E, PP) (DP, PE, FORP) (EOP, FORP, Oa E) (DP, PE, FORP) (EOP, Oa E, Tr. Eq.) (EOP, FORP, PP) (DP, PE, Tr. Eq.) (PE, PP, FORP) (FORP, Oa E, Tr. Eq.) (PE, PP, Oa E) (PE, Oa E, Tr. Eq.) (PE, Oa E, FORP) (PE, PP, Tr. Eq.) (PP, Oa E, Tr. Eq.) (PP, FORP, Tr. Eq.) The proofs for all other combinations of three fairness definitions can be found in Appendix B, and the results are summarized in Table 4. 3.3.1 P%-Rule The p%-rule is investigated separately from the more strict fairness definitions. In order to determine the compatibility of the p%-rule with two other fairness definitions, we introduce the following two inequalities in the set of linear equations: TPb+FPb p 100, TPb+FPb TPa+FPa p 100. The combination of these equations constitute the p%-rule via the confusion matrix. The calculations for all combinations with two other fairness definitions can be found in Appendix C. The results can be divided into three categories: solvable for all p, an upper bound on p, or for p = 1 there is a specific value for certain variables. This property is noted for each combination in Table 5. Table 5: The compatibility of the p%-rule with two fairness definitions. p1 Natural bound2 Set value for a variable3 (EOP, PE) (EOP, Tr. Eq.) (EOP, PP) (PE, FORP) (FORP, Oa E) (PE, PP) (EOP, FORP) (PE, Tr. Eq.) (FORP, Tr. Eq.) (PP, Tr. Eq.) (EOP, Oa E) (PP, Oa E) (Oa E, Tr. Eq.) Maximal Combinations of Fairness Definitions 3.3.2 Extending to Multiple Sensitive Groups Like for combining two fairness definitions, the calculations can be conveniently extended to multiple sensitive groups. The combinations (EOP, PP, Tr. Eq.) and (PE, FORP, Tr. Eq.) both contain two free variables; their extension to multiple sensitive groups follows the method discussed in Section 3.2.2. Other possible combinations have only one free variable, belonging to one sensitive group. The confusion matrices of other groups are defined by the base rates and the one free variables. This is repeated for every disjoint sensitive group. 3.4 Combining Four Fairness Definitions There are only four free variables across the confusion matrices for two sensitive groups, due to the inherent constraints in a confusion matrix as noted in Eq. (1). Therefore, combinations of four independent fairness notions will not be possible. In other words: for a combination of four fairness definitions to be possible, at least one of the constraints should mathematically follow from the combination of other constraints in order for the values in the confusion matrix to have a non-zero Lebesgue measure. Proposition 3. Out of demographic parity, equal opportunity, predictive equality, predictive parity, false omission rate parity, overall accuracy equality and treatment equality no set of four fairness definitions is possible (according to Definition 1). Proof outline. If a set of four fairness definitions were possible, then all subsets of three out of those four definitions should be possible. Based on Table 4, we can exhaustively verify that this is not the case: no set of size four exists for which all its size three subsets are possible. Thus, no possible combination of four fairness definitions exist. 3.4.1 P%-Rule The compatibility of three fairness definitions with the p%-rule can be found by combining the previous constraints of the combination of the p%-rule with two fairness definitions. Simply put, the intersection of the possible values of p of all the combinations with two of the relevant fairness definitions will constitute the range of possible values for p when combined with three fairness definitions. It is not possible to satisfy three fairness definitions and the p%-rule for every possible value of p. This can be deduced from the incompatibility that no three fairness definitions can be extended with demographic parity, which can be seen as the strictest form of the p%-rule. 3.5 Integration of the Results Combining the propositions above, we now state the main result of this paper. Theorem 1. Twelve sets form a maximal combination of fairness definitions that are possible according to Definition 1, out of demographic parity, equal opportunity, predictive equality, predictive parity, false omission rate parity, overall accuracy equality and treatment equality. We call these twelve combinations the maximal fairness notions. Seven of these sets are pairwise combinations and five are a combination of three fairness definitions. All subsets from these twelve sets are also possible. Defrance & De Bie Proof outline. Figure 1 visualizes the sets of possible combinations out of the results in Propositions 1, 2, and 3. This tree of possible sets contains twelve leaf nodes, meaning that these twelve nodes are the maximal sets of possible fairness definitions. The nodes representing a maximal set of fairness definitions are marked with an asterisk(*). A relation is found between the possible combinations. This relation is shown in Table 6. It shows how often each fairness definition is possible to combine with two others, is impossible to combine as it requires a trivial 2-dimensional range for base rates, or requires a trivial range either for the variables or a trivial 2-dimensional range for the base rates. Table 6: The prevalence of each fairness definition when combined with two others as the different characteristics of the outcome. Definition Count possible combinations Count requiring equal base rates Count trivial range for variables or base rates Demographic Parity 1 3 11 Equal Opportunity 3 2 10 Predictive Equality 3 2 10 Predictive Parity 2 6 7 False Omission Rate Parity 2 6 7 Overall accuracy Equality 2 2 11 Treatment Equality 2 3 10 Table 6 reveals a symmetry between equal opportunity and predictive equality and also between predictive parity and false omission rate parity. This symmetry is due to their mathematical symmetry, as formalized by the following proposition. Proposition 4. Given any possible combination of fairness definitions, another possible combination can be obtained by swapping the semantics of the classes, i.e. by renaming the positive class as the negative and vice versa. Proof outline. When visualization fairness definitions on the confusion matrix, swapping the positive and negative classes is equivalent to a point inversion at the center of the confusion matrices. Figure 5 illustrates the point inversions of all possible combinations of three fairness definitions, revealing the symmetric relation between them. A formal proof follows from the symmetry relations between the individual fairness definitions. Switching positives and negatives in the constraint of equal opportunity results in the constraint for predictive equality, and vice versa. The same relation holds between predictive parity and false omission rate parity. However, demographic parity becomes an equivalent constraint to the original definition. The definitions of overall accuracy equality and treatment equality remain unaffected by switching the positive and negative classes. Proposition 4 explains the symmetry in Table 6 of equal opportunity with predictive equality and of predictive parity with false omission rate parity. If for every possible combination their inversion is also a possible combination, then for every possible combination Maximal Combinations of Fairness Definitions Figure 5: Point inversion of the possible combinations of three fairness definitions. with predictive parity there will be a possible combination with false omission rate parity. The relation between equal opportunity and predictive equality is analogous. Figure 5 shows that three, among the five maximal combinations of three fairness definitions, treat the positive and negative classes equally; they remain unaltered by swapping the role of the positive and negative class. These three combinations are (DP, EOP, FPP), (EOP, PP, Oa E), and (PP, PE, Oa E). The combinations (EOP, PP, Tr. Eq.) and (FPP, PE, Tr. Eq.) treat the classes differently, and represent each other s symmetrical analogue. Finally, both Figure 1 and Table 6 indicate that possible combinations with demographic parity are most rare. Demographic parity does not relate to the base rates of the groups as it is independent from the ground truth. Therefore, it can only be combined with definitions that relate directly to the base rates such as equal opportunity and predictive parity. 3.6 Relation to Existing Impossibility Theorems Our work is congruent with previous impossibility theorems. The work of Chouldechova (2016) states, it is not possible to have equal positive prediction values, false negative rates, and false positive rates between groups for different the base rates. We call these definitions predictive parity, predictive equality, and equalised odds. As can be seen in Figure 1, our work comes to the same conclusion as the work of Chouldechova. Another impossibility theorem of Kleinberg et al. (2016) is centred on scoring instead of binary prediction. The incompatible fairness definitions for risk scoring can be mapped onto binary prediction. Calibration is seen as predictive parity, balance for the negative class as predictive equality, and balance for the positive class as equal opportunity. This matches what Chouldechova (2016) proved and our results. Barocas et al. (2019) state that demographic parity is not compatible with equalised odds. Initially, this seems to contradict our results. However, Barocas et al. introduce the constraint on combining fairness definitions that predictions cannot be independent from the ground truth. As pointed out in Remark 3, this could be a desirable constraint for a definition of compatibility, but was not adopted in Def. 1. Def. 1 focusses on the possibility of finding a confusion matrix configuration satisfying multiple fairness definitions. Sec. 4 concerns the performance constrained seen when enforcing multiple fairness definitions. Defrance & De Bie 4. Analysis of The Fairness Combinations The following section discusses the performance and characteristics of the combinations of fairness definitions. We first discuss the maximal accuracies each group can achieve when enforcing multiple fairness definitions. In the last section we discuss the performance in more detail. We discuss the type of constraint that is posed on the confusion matrix in order to provide some insight on the resulting behaviour of the system. 4.1 Performance Bounds of Combinations of Fairness Definitions Enforcing a set of fairness definitions imposes extra constraints on the classifier. This might affect the performance on the model. We discuss the effects each set of fairness definitions will have on the accuracy for each group. Below, for the sake of the argument, we take the positive class to be preferable over the negative class. The calculations from Sections 3.2 and 3.3 can be reused to determine the minimal and maximal accuracy a predictor can achieve. This performance is dependent on the difference in base rates and for some combinations on the absolute values of the base rate. This relation between the accuracies of both groups is visualized for some combinations of fairness definitions in Figure 6. Figure 6 shows that the behaviour of the accuracies vary greatly depending on the combinations of fairness definitions. Such as for Figure 6a, the combination of demographic parity with predictive parity, where for large differences between the base rates of groups A and B, the performance for group A is greatly limited. Contrarily, the combination of equal opportunity and predictive equality (Figure 6b) can achieve high performance for both groups regardless of the difference in base rates. The formulas to calculate the minimal and maximal accuracy for each feasible combination can be found in Table 7. In calculations of the maximal accuracy, the variable εx either refers to the false positive or false negative rate within a specific group. For the minimal accuracy it signifies the true positive and true negative rate. From Table 7, we can see that the combinations with demographic do not have accuracies within a the range [0, 1]. This range is dependant on the base rates. This means that a perfect predictor is impossible while enforcing a set of fairness definitions that contains demographic parity. Example of Performance Influence with The COMPAS Dataset We briefly show how these performance bounds have an effect on a real-world example. The COMPAS dataset (Angwin et al., 2016) is used for this example as this was used in the introduction. It has been argued that the COMPAS dataset is not appropriate to use in a benchmarking setting (Fabris et al., 2022). However, this example is fictional and the only information used in the subsequent calculations are the base rates. We limit ourselves to two sensitive groups, namely Caucasian and African American. The base rates are 0.43 and 0.54 respectively for the Caucasian and African American group. The COMPAS case is peculiar as an individual prefers the negative prediction. Common fairness use cases such as loan applications or job applications have a preferred positive prediction. Therefore, the advantaged group is the Caucasian as it has a lower base rate. Maximal Combinations of Fairness Definitions Table 7: The minimal and maximal accuracy of each possible combination of fairness definitions except for the combination of (PP, FORP) and (PP, FORP, Oa E). εx denotes the false positive or false positive for a specific group in the maximal accuracy calculations and the true positive or true negative rates in the minimal accuracy calculations. The value of εx is thus constrained by the properties of the set of combinations. Specific ranges can be found in Appendix A and B. Minimal accuracy Maximal Accuracy Advantaged Disadvantaged Advantaged Disadvantaged Pairwise Combinations (DP, EOP) x y + 1+y 2x 1 y ε1 + ε2 ε1 + ε2 1 + y x 1+y 2x 1 y ε1 ε2 1 ε1 ε2 (DP, PE) y x + 2x y y ε2 + ε1 ε1 + ε2 1 + x y y 2x y ε2 ε1 1 ε1 ε2 (DP, PP) x y + ε1 + ε2 ε1 + ε2 1+x y ε1 ε2 1 ε1 ε2 (DP, FORP) y x + ε1 + ε2 ε1 + ε2 1 x+y ε1 ε2 1 ε1 ε2 (DP, Oa E) 0 0 1 y x (DP, Tr. Eq.) ε1 + ε2 + (y x)(1 ε1 ε2) 2 y 1 ε2+ε1 ε1 + ε2 1 ε1 ε2 (ε1+ε2)(x y) ε2 ε1 1 ε1 ε2 (EOP, PE) 1 x 1 yε1 + x yε2 ε1 + ε2 1 1 x yε2 1 ε1 ε2 (EOP, FORP) 1 x 1 y(ε1 + ε2) ε1 + ε2 1 x 1 y(1 ε1 ε2) 1 ε1 ε2 (EOP, Oa E) 1 x y xε2 1 y y xε1 1 x y xε2 1 y y xε1 1 ε2 1 ε2 (PE, PP) x y(ε1 + ε2) ε1 + ε2 x y(1 ε1 ε2) 1 ε1 ε2 (PE, Oa E) x x yε2 y x yε1 x x yε2 y x yε1 1 ε 1 ε (PP, Oa E) ε1 + ε2 ε1 + ε2 1 ε1 ε2 1 ε1 ε2 (FORP, Oa E) ε1 + ε2 ε1 + ε2 1 ε1 ε2 1 ε1 ε2 (Oa E, Tr. Eq.) ε1 + ε2 ε1 + ε2 1 ε1 ε2 1 ε1 ε2 Combination of three fairness definitions (DP, EOP, PE) x + 1 2 x 1 y ε y + 1 2 y 1 y ε 1 x 1 2 x 1 y ε 1 y 1 2 y (EOP, PE, Oa E) 1 yε 1 yε 1 ε (EOP, PP, Tr. Eq.) x y 1 y + 1 x 1 y(ε1 + ε2) ε1 + ε2 1 1 x 1 yε1 1 x 1 yε2 1 ε1 ε2 (PE, FORP, Tr. Eq.) y x y(ε1+ε2) ε1 + ε2 1 x yε2 1 ε1 ε2 Defrance & De Bie (a) (DP, PP) (b) (EOP, PE) (c) (EOP, FORP) (d) (DP, OEP) Figure 6: The maximal accuracies for groups A and B. We assume the base rate of group A = 0.7 and FNa = 2 FPa (Reducing the complexity to allow for visualizations). White regions in the graph denotes impossible results. Performance borders for group B are drawn for the values 0.2, 0.4, 0.6 and 0.8. We posit that the most interesting fairness definitions for this use case are Predictive Equality and Predictive Parity as both these definitions aim to impose some type of parity related to the false positives. We focus on these false positive as these are the worst possible outcome for an individual. Figure 7 shows some examples of the performance characteristics when enforcing a combination of fairness definitions. The combination of Predictive Equality and Predictive Parity could seem opportune in such a system. However, Figure 7d shows that the performance discrepancy between groups for these base rates is large, which can be considered undesirable. The performance of the fairness-unaware classifier often provides an upper bound (Liu & Vicente, 2022) for the performance of a fairness-aware classifier, providing a likely accuracy region of the fairness-aware classifier. For lower accuracy levels, a significant performance difference can be observed for the combinations DP & PP (Fig. 7b) and PP & Tr.Eq. (Fig. 7f). Therefore, these combinations can be seen as undesirable for lower accuracies. The other three combinations of fairness definitions shown in Figure 7 exhibit more preferable properties. Figure 7e shows that the combination PE & Tr.Eq. has a large breadth of possible combinations, meaning that a desirable solution could be found. The Maximal Combinations of Fairness Definitions (a) Enforcing DP and PE (b) Enforcing DP and PP (c) Enforcing EOP and PE (d) Enforcing PE and PP (e) Enforcing PE and Tr. Eq. (f) Enforcing PP and Tr. Eq. Figure 7: Possible range of results when enforcing a set of fairness constraints for the COMPAS dataset for the groups of Caucasian and African American individuals. combinations of DP & PE and EOP & PE, as shown in Figures 7a and 7c respectively, have a fairly wide range of solutions and require smaller differences in the accuracies of the groups compared to the other portrayed combinations of fairness definitions. From a performance perspective this is desirable as there is no risk of sizeable performance differences between sensitive groups when a model satisfies these combinations of fairness definitions. Remark 4. The desired combination of fairness definitions should not be determined based on performance. A choice of fairness definitions is an ethical choice that has to be made independently. However, negative influences on the performance of the model might eliminate certain combinations from consideration. 4.2 Behaviour of Feasible Combinations of Fairness Definitions The calculations in Sections 3.2 and 3.3 allow to group the sets of possible combinations based on the resulting constraints that are imposed on the confusion matrices. Five distinct behaviour types and their correspondent sets are noted in Table 8. An example for each of the behaviour types identified in Table 8 is provided in Tables 9 and 9c in the form of the resulting confusion matrices when enforcing such a set of fairness definitions. We discuss these behaviour in the remainder of this section. Lower bound on one variable A lower bound is imposed on one variable in the confusion matrix. This lower bound is dependent on the base rates of the groups. The lower bound is primarily dependent on the difference in base rates when combining two fairness definitions. An example of such behaviour in the confusion matrix can be found in Table 9a. Defrance & De Bie Table 8: The different feasible combinations of fairness definitions sorted on the behaviour they exhibit in the confusion matrices Behaviour type Combinations of fairness definitions Lower bound on one variable (DP, PP) (DP, FORP) (DP, Oa E) (DP, EOP, PE) Trade-off between errors (DP, EOP) (DP, PE) (PP, Oa E) (FORP, Oa E) (EOP, Oa E) (PE, Oa E) Factor on the accuracy (EOP, FORP) (PE, PP) Factor on the error (EOP, PP, Tr.Eq.) (PE, FORP, Tr. Eq.) (EOP, PE, Oa E) (EOP, PE) No effect (Oa E, Tr. Eq.) It shows that there will either be false negatives for the one group or for the other group as it is impossible to satisfy both y x ε1 = 0 and ε1 = 0 if the base rates differ. The combination of demographic parity, equalised opportunity, and predictive equality also contains a lower bound, though on the false positives. Unlike the combinations with two fairness definitions, the relation between the base rates and the lower bound is more complicated as can be seen in Table 9b. Trade-off between errors Two errors of the other group contribute to one of the variables in the confusion matrix. In this case, the value from one errors constrains the possible range of values for the other error. An example is shown in Table 9c for the combination of predictive parity and overall accuracy equality. The presence of x y in the equation means that the difference in base rates will influence the possible ranges of these errors. Factor on the TP/TN True positives or true negatives are related through a multiplicative factor, derived from the base rates. An example of this relation on the true negatives is shown in Table 9d. Unless the base rate are equal, this causes that no perfect accuracy can be achieved for one of the groups. This can also be seen in Table 7, as the combinations EOP & FORP and PE & PP have that factor in their maximal accuracy. Factor on the error This is closely related to the previous behaviour, however the factor is on the errors. Although similar, this characteristic does not impose the same limitation on the accuracies of the groups. If one group has perfect accuracy, then the other group will too, as all errors will be zero. An example of this can be found in Table 9e. No effect Only the combination of overall accuracy equality and treatment equality have no special properties are present on the confusion matrix, as can be seen in Table 9f. 5. Conclusion We investigated which and how many fairness definitions can be combined and imposed simultaneously for the simple setting of binary classification. We find that out of seven fairness definitions, namely demographic parity, equal opportunity, predictive equality, predictive parity, false omission rate parity, overall accuracy equality and treatment equality, in total twelve maximal combinations are possible. Five of these maximal combinations Maximal Combinations of Fairness Definitions (a) Lower bound on one variable: Demographic parity and predictive parity Group A with base rate: 1 x Predicted Pos. Neg. True Pos. 1 y ε1 y x + ε1 Neg. ε2 x ε2 Group B with base rate: 1 y Predicted Pos. Neg. True Pos. 1 y ε1 ε1 Neg. ε2 y ε2 (b) Lower bound on one variable: Demographic parity, equalised odds and predictive equality. Group A with base rate: 1 x Predicted Pos. Neg. True Pos. 1 x 1 x 1 yε 1 x 1 yε Neg. x x 1 yε x 1 yε Group B with base rate: 1 y Predicted Pos. Neg. True Pos. 1 y ε ε Neg. y y 1 yε y 1 yε (c) Trade-off between errors: Predictive parity and overall accuracy equality Group A with base rate: 1 x Predicted Pos. Neg. True Pos. ε1 y x ε1 ε2 + ε1 1 x ε1 y x ε1 ε2 ε1 Neg. ε1 x ε1 Group B with base rate: 1 y Predicted Pos. Neg. True Pos. ε2 y x ε1 ε2 + ε2 1 y ε2 y x ε1 ε2 ε2 Neg. ε2 y ε2 (d) Factor on the TP/TN: Equal opportunity and false omission rate parity. Group A with base rate: 1 x Predicted Pos. Neg. True Pos. 1 x 1 x 1 yε1 1 x 1 yε1 1 y(y ε2) 1 x Group B with base rate: 1 y Predicted Pos. Neg. True Pos. 1 y ε1 ε1 Neg. ε2 y ε2 (e) Factor on the error: Equal opportunity and predictive equality. Group A with base rate: 1 x Predicted Pos. Neg. True Pos. 1 x 1 x 1 yε1 1 x 1 yε1 Neg. x yε2 x x Group B with base rate: 1 y Predicted Pos. Neg. True Pos. 1 y ε1 ε1 Neg. ε2 y ε2 (f) No effect: Overall accuracy equality and treatment equality. Group A with base rate: 1 x Predicted Pos. Neg. True Pos. 1 x ε1 ε1 Neg. ε2 x ε2 Group B with base rate: 1 y Predicted Pos. Neg. True Pos. 1 y ε1 ε1 Neg. ε2 y ε2 Table 9: Examples of the different behaviour types identified in Table 8 and their corresponding confusion matrices. 1 x, 1 y are the base rates of groups A and B respectively. εx denotes an error of some type, and functions as a variable. Defrance & De Bie consist of three fairness definitions, while seven of them consist of just two fairness definitions. Each of these sets could be regarded as maximal fairness notions, which means that it is not possible to impose any further of the discussed fairness constraints without the problem becoming infeasible. Evidently, all subsets of these twelve maximal sets are also possible combinations of fairness definitions. An overview of the possible combinations, maximal as well as non-maximal, and their subset relations, is given in Figure 1. Our results confirm and extend related work, such as the theorem colloquially referred to as the fairness impossibility theorem. We consider the five maximal sets of three fairness definitions as particularly interesting. Of these five, three treat the positive and negative classes symmetrically, in that they remain unaltered after swapping the class labels. The remaining two are each other s symmetrical analogue: one of these two sets of fairness definitions is equivalent to the other set after swapping the class labels. These properties are shown in Figure 5. They can provide initial guidance to determine which set of fairness notions is most appropriate in a given setting: if the costs of different types of misclassifications are very different, one of the sets of fairness notions that are not invariant with respect to a label swap might be more appropriate. Furthermore, we investigated the properties of the possible combinations. This included the effects they have on the accuracy of the classifier for each of the groups. Those results showed that certain combinations of fairness definitions are impractical if the difference between the base rates are too large. Additionally, we studied the type of effect the possible combinations had on the confusion matrix and how they constrain the confusion matrices. Five behaviour types could be discerned. An important next step building on these results would be to develop methods that are capable of imposing these (maximal) sets of fairness definitions. Subsequently, it would be interesting to investigate the impact of imposing (maximal) sets of fairness definitions on the learning capabilities (e.g. learning rate and convergence rate) of machine learning methods. A separate line of further work would be towards including other fairness definitions, in addition to the seven definitions included in this work. Moreover, we consider extensions towards other problems besides binary classification as fruitful avenues for further research. Finally, and perhaps most importantly, the meaning of the maximal fairness notions derived in the present paper needs to be better understood from an ethical and/or legal point of view. This is an important aspect of research in fairness in AI as this translation to real world concepts is a pre-requisite for wide-spread adoption. Acknowledgments The research leading to these results has received funding from the Special Research Fund (BOF) of Ghent University (BOF20/IBF/117), from the European Research Council under the European Union s Seventh Framework Programme (FP7/2007-2013) (ERC Grant Agreement no. 615517), and under the European Union s Horizon 2020 research and innovation programme (ERC Grant Agreement no. 963924), from the Flemish Government under the Onderzoeksprogramma Artifici ele Intelligentie (AI) Vlaanderen programme, and from the FWO (project no. G0F9816N, 3G042220, G073924N). Maximal Combinations of Fairness Definitions Appendix A. Combining two definitions A.1 Equal Opportunity and Demographic Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa TPa+FNa = TPb TPb+FNb TPa+FPa TPa+FPa+FNa+TNa = TPb+FPb TPb+FPb+FNb+TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa 1 x = TPb 1 y TPa + FPa = TPb + FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = TPb + FPb 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = (1 1 x 1 y)TPb + FPb FNa = 1 x 1 x 1 y TPb TNa = x x y 1 y TPb FPb FNb = 1 y TPb TNb = y FPb TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb A.1.1 Applying Inequalities 1 y TPb 1 x 1 x 1 y TPb 1 x 1 y TPb FPb x x y 1 y TPb FPb x 0 1 y TPb 1 y TPb 1 y 0 y FPb y FPb y 0 1 x 1 y TPb 1 x 1 y TPb 1 x 1 y TPb + FPb x y 1 y TPb + FPb x 1 1 y TPb 1 1 x 1 y TPb 0 FPb x + x+y 1 y TPb x+y 1 y TPb FPb TPb 1 y TPb 0 FPb y FPb 0 0 1 x 1 y 1 1 y TPb 1 y x 1 y FPb TPb FPb + x y TPb FPb + x y Defrance & De Bie A.2 Demographic Parity and Predictive Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa + FPa = TPb + FPb FPa = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb TPa = TPb + (1 x FNa = 1 x y x y FPb TPb TNa = x x y FPb FNb = 1 y TPb TNb = y FPb FPa = x y FPb TPa = y x y FPb + TPb A.2.1 Added Constraints through Inequalities FPb 1 x y x y FPb + TPb A.3 Demographic Parity and Predictive Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa FPb = TPb FPa TPa + FPa = TPb + FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb TPb FPa FPb + FPa = TPb + FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb FPb+TPb FPb FPa = TPb + FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb FPa = FPb FPb + TPb = 0 Maximal Combinations of Fairness Definitions FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb FNa = 1 x FPa TNa = x FPa FNb = 1 y TNb = y TPa = FPa FPb = TPb = 0 A.3.1 Added Constraints through Inequalities ( 1 x TPb x FPb A.4 Demographic Parity and False Omission Rate Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa TNb = FNb TNa TPa + FPa = TPb + FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb TNa TNb 1 FNa TNa = 1 FNb TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb TNa TNb FNa + TNa = FNb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb TNa TNb FNb TNa TNb + TNa = FNb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb TNa TNb FNb+TNb TNb TNa = FNb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = TNb FNb = TNb = 0 FNa = FNb TNa Defrance & De Bie TPa = 1 x FNa FPa = x FNa TPb = 1 y FPb = y FNb = TNb = 0 FNa = TNa TPa = 1 x FNb FPa = x TNb TPb = 1 y FNb FPb = y TNb TNa = TNb FNa = FNb A.4.1 Added Constraints through Inequalities ( FNb 1 x TNb x A.5 Demographic Parity and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa + FPa = TPb + FPb TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa TPb = FPb FPa TPa TPb = TNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNb TNa = FPb FPa TPa TPb = FPb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y y FPb x + FPa = FPb FPa TPa TPb = FPb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y y x = 2FPb 2FPa TPa TPb = FPb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPb FPa = y x 2 TPa TPb = y x FNa = 1 x TPb y x 2 TNa = x FPb x y 2 FNb = 1 y TPb TNb = y FPb FPa = FPb + x y 2 TPa = TPb + y x Maximal Combinations of Fairness Definitions A.5.1 Added Constraints through Inequalities A.6 Demographic Parity and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa + FPa = TPb + FPb FNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FNa FPb FNb 1 x FNa + FNa FPb FNb = 1 y FNb + FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FNa FPb FNb FPb FNb FNb FNa = x y + FPb FNb TPa = 1 x FNb( x y FPb FNb + 1) TNa = x FPb( x y FPb FNb + 1) TPb = 1 y FNb TNb = y FPb FNa = FNb( x y FPb FNb + 1) FPa = FPb( x y FPb FNb + 1) TPa = 1 x FNa TNa = x FNa TPb = 1 y FNb TNb = y FNb FPb = FNb x = y FPa = FNa A.6.1 Added Constraints through Inequalities x y FPb FNb 1 1 x FNb x FPb Defrance & De Bie A.7 Equalised Odds TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa FPa+TNa = FPb FPb+TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa FNa = 1 x 1 x 1 y TPb TNa = x x y FPb FNb = 1 y TPb TNb = y FPb TPa = 1 x 1 y TPb FPa = x A.8 Equal Opportunity and Predictive Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TPa TPa+FPa = TPb TPb+FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TPa FPb = TPb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb 1 x 1 y TPb FPb = TPb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb 1 x 1 y FPb = FPa TPb = 0 FNa = 1 x 1 x 1 y TPb TNa = x 1 x 1 y FPb FNb = 1 y TPb TNb = y FPb TPa = 1 x 1 y TPb FPa = 1 x FNa = 1 x TNa = x FPa FNb = 1 y TNb = y FPb TPa = 0 TPb = 0 A.8.1 Added Constraint through Inequalities Maximal Combinations of Fairness Definitions A.9 Equal Opportunity and False Omission Rate Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FNa FNa+TNa = FNb FNb+TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FNa TNb = FNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y 1 x FNa = 1 x 1 y(1 y FNb) FNa TNb = FNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb 1 x 1 y FNb TNb = FNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb 1 x 1 y TNb = TNa FNb = 0 TPa = 1 x 1 x 1 y FNb FPa = x 1 x 1 y TNb TPb = 1 y FNb FPb = y TNb FNa = 1 x 1 y FNb TNa = 1 x TPa = 1 x FPa = x TNa TPb = 1 y FPb = y TNb FNa = 0 FNb = 0 A.9.1 Added Constraint through Inequalities Defrance & De Bie A.10 Equal Opportunity and Overall accuracy equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb 1 x 1 y TPb TPb = TNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb y x 1 y TPb = TNb TNa FNa = 1 x 1 x y x(TNb TNa) FPa = x TNa FNb = 1 y 1 y y x(TNb TNa) FPb = y TNb TPa = 1 x y x(TNb TNa) y x(TNb TNa) A.10.1 Added Constraints through Inequalities y > x TNa TNb y x TNb TNa x > y TNb TNa y x TNb TNa A.11 Equal Opportunity and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb 1 x 1 y FNb FPa = FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = 1 x 1 y FPb FNb = 0 Maximal Combinations of Fairness Definitions TPa = 1 x 1 x 1 y FNb TNa = x 1 x 1 y FPb TPb = 1 y FNb TNb = y FPb FNa = 1 x 1 y FNb FPa = 1 x TPa = 1 x TNa = x FPa TPb = 1 y TNb = y FPb FNa = 0 FNb = 0 A.11.1 Added Constraints through Inequalities 1 y 1 xx FPb A.12 Predictive Equality and Predictive Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb TPa FPb = TPb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb TPa FPb = TPb x FNa = 1 x TPa TNa = x x y FPb FNb = 1 y TPb TNb = y FPb FPa = x y FPb TPa = x y TPb FPb = 0 A.12.1 Added Constraints through Inequalities y x(1 x) TPb A.13 Predictive Equality and False Omission Rate Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb FNa TNb = FNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y x TNa = x FNa TNb = FNb TNa Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb FNa TNb = FNb x TPa = 1 x FNa FPa = x x y TNb TPb = 1 y FNb FPb = y TNb TNa = x y TNb FNa = x y FNb TNb = 0 A.13.1 Added Constraints through Inequalities y x(1 x) FNb A.14 Predictive Equality and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa + x y TNb = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb = TPb TPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNb = y x y(TPb TPa) y y x y(TPb TPa) FNa = 1 x TPa FPa = x x x y(TPb TPa) FNb = 1 y TPb FPb = y y x y(TPb TPa) TNb = y x y(TPb TPa) TNa = x x y(TPb TPa) A.14.1 Added Constraints through Inequalities x > y TPa TPb x y TPb TPa x < y TPa TPb x y TPb TPa Maximal Combinations of Fairness Definitions A.15 Predictive Equality and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb FNa FPb = FNb x TPa = 1 x FNa TNa = x x y FPb TPb = 1 y FNb TNb = y FPb FPa = x y FPb FNa = x y FNb FPb = 0 A.15.1 Added Constraints through Inequalities y x(1 x) FNb A.16 Predictive Parity and False Omission Rate Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb FNa = FNb TNa A.17 Predictive Parity and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa FPb = TPb FPa TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb TPb FPa FPb + x FPa = TPb + y FPb Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb FPa FPb FPb TPb = y x + FPa FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb TPb = FPb y x FPa FPb + FPb (FPa = FPb x = y) FNa = 1 x FPa y x FPa FPb FPa TNa = x FPa FNb = 1 y FPb y x FPa FPb FPb TNb = y FPb TPb = FPb y x FPa FPb + FPb TPa = FPa y x FPa FPb + FPa FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb x = y A.17.1 Added Constraints through Inequalities 1 y 1 x FPa FPb 1 y FPa 1 x FPa FPb FPa 1 y+FPb 1 x+FPb FPb 1 y 1 x FPa FPb 1 y FPa 1 x FPa FPb FPa 1 y+FPb 1 x+FPb FPb A.18 Predictive Parity and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa FPb = TPb FPa FNa FPb = FNb FPa Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb FPa FPb (1 x FNb FPa FPb ) FPb = (1 y FNb) FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb FPa FPb FPb x FPb FNb FPa = FPa y FPa FNb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb FPa FPb FPb x FPb = FPa y FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb FPa FPb FPa = 1 x TPa = 1 x 1 x 1 y FNb TNa = x 1 x 1 y FPb TPb = 1 y FNb TNb = y FPb FNa = 1 x 1 y FNb FPa = 1 x A.18.1 Added Constraints through Inequalities 1 y 1 xx FPb A.19 False Omission Rate Parity and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa TNb = FNb TNa TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb TNa TNb x + FNa TNa = y + FNb TNb Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb TNa TNb TNb FNb TNb TNa = x y + TNb FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = TNb x y TNb FNb + TNb (TNb = FNb x = y) FNa = FNb TNa TPa = 1 x FNb x y TNb FNb FNb FPa = x TNb x y TNb FNb TNb TPb = 1 y FNb FPb = y TNb TNa = TNb x y TNb FNb + TNb FNa = FNb x y TNb FNb + FNb TPa = 1 x TNa FPa = x TNa TPb = 1 y FNb FPb = y FNb TNb = FNb x = y FNa = TNa A.19.1 Added Constraints through Inequalities TNb FNb y x x TNb y TNb TNb FNb 1 x FNb 1 y FNb FNb TNb FNb y x x TNb y TNb TNb FNb 1 x FNb 1 y FNb FNb A.20 False Omission Rate Parity and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa TNb = FNb TNa FNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = FNb TNa TNb FNa = FNb FPa Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNb FPa FPb = FNb TNa TNb FNa = FNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa FPb = TNb FPa FNb = 0 FNa = FNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa(y TNb) = TNb(x TNa) FNb = 0 FNa = FNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb FNb = 0 FNa = FNb TNa TPa = 1 x x y FNb FPa = x x y TNb TPb = 1 y FNb FPb = y TNb TNa = x y TNb FNa = x TPa = 1 x TNa = x FPa TPb = 1 y TNb = y FPb FNb = 0 FNa = 0 A.20.1 Added Constraints through Inequalities y x(1 x) TNb A.21 Overall accuracy Equality and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa + FNa = FPb + FNb FNa = FNb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa + FNb FPa FPb = FPb + FNb FNa = FNb FPa Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa FPb+FNb FPb = FPb + FNb FNa = FNb FPa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb FPb = FNb = 0 FNa = FNb FPa TPa = 1 x FNb TNa = x FPb TPb = 1 y FNb TNb = y FPb FPa = FPb FNa = FNb TPa = 1 x TNa = x FPa TPb = 1 y TNb = y FPb = FNb = 0 FNa = 0 A.21.1 Added Constraints through Inequalities ( x FPb 1 x FNb Appendix B. Combining Three Definitions B.1 Demographic Parity and Equalised Odds TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb TPa = y x y FPb + TPb FPa = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y FPb x y FPb = x y 1 y TPb + FPb 1 x 1 y TPb = y x y FPb + TPb Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y FPb = x y 1 y TPb 1 x 1+y 1 y TPb = y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y FPb FPb = y 1 y TPb x = y y x 1 y TPb = y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y FPb FPb = y 1 y TPb y x 1 y TPb = y x y y 1 y TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb x = y 0 = 0 FNa = 1 x 1 x 1 y TPb TNa = x x 1 y TPb FNb = 1 y TPb TNb = y y 1 y TPb TPa = 1 x 1 y TPb FPa = x 1 y TPb FPb = y 1 y TPb TPb = TPb FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb x = y 0 = 0 B.2 Demographic Parity, Equal Opportunity and Predictive Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb TPa = TPb FPa = FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPa TPa = TPb FPa = FPb Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb x y 1 y TPb = 0 TPa = TPb FPa = FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb x y = 0 TPb = 0 TPa = TPb FPa = FPb FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb x = y TPa = TPb TPa = TPb FPa = FPb FNa = 1 x TNa = x FPb FNb = 1 y TNb = y FPb TPb = 0 TPa = 0 TPa = 0 FPa = FPb B.3 Demographic Parity, Equal Opportunity and False Omission Rate Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb TNa = TNb FNa = FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb FPa = x + y FPb TPa = 1 + x + 1 y TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb TPa = x + y + TPb x + y + TPb = 1 x 1 y TPb x y + FPb = x y 1 y TPb + FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb TPa = x + y + TPb x + y = y x 1 y TPb x y = x y Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb TPa = x + y + TPb 1 = 1 1 y TPb x = y 1 = 1 1 y TPb x = y FNa = 0 TNa = y FPb FNb = 0 TNb = y FPb FPa = x y + FPb TPa = 1 x TPb = 1 y FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb FPa = FPb TPa = TPb x = y B.4 Demographic Parity, Equal Opportunity and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb FPa = FPb + x y 2 TPa = TPb + y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb x y 1 y TPb + FPb = FPb + x y 2 1 x 1 y TPb = TPb + y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb x y 1 y TPb = x y 2 y x 1 y TPb = y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb x y 1 y TPb = x y Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb TPb = 1 y 2 TNa = x x y 2 FPb FNb = 1 y 2 TNb = y FPb TPa = 1 x 2 FPa = x y 2 + FPb TPb = 1 y FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb x = y B.5 Demographic Parity, Equal Opportunity and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y 1 y TPb + FPb FNa = FNb( x y FPb FNb + 1) FPa = FPb( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y 1 y (1 y FNb) + FPb FNa = FNb( x y FPb FNb + 1) FPa = FPb( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb + FPb 1 x 1 y FNb = FNb( x y FPb FNb + 1) 1 y FNb + FPb = FPb( x y FPb FNb + 1) Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb + FPb 1 x 1 y = ( x y FPb FNb + 1) FNb = 0 1 y FNb = FPb( x y FPb FNb ) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb + FPb 1 1 y = 1 FPb FNb FNb = 0 x = y (x y)(1 1 1 y FNb) = FPb x y FPb FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb + FPb 1 + y = FPb FNb FNb = 0 x = y 1 1 1 y FNb = FPb FPb FNb x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb + FPb 1 + y = FPb FNb FNb = 0 x = y FPb FNb 1 1 y(FPb FNb)FNb = FPb x = y Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb + FPb 1 + y = FPb FNb FNb = 0 x = y 1 1 1 y(FPb FNb) = 0 x = y FNb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb + FPb 1 + y = FPb FNb FNb = 0 x = y FPb FNb = 1 + y x = y FNb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb + FPb FPb = 1 + y + FNb FNb = 0 x = y TPa = 1 x 1 x 1 y FNb TNa = y + x y 1 y FNb FNb = 1 y TNb = y FNa = 1 x 1 y FNb FPa = x y x y 1 y FNb FPb = TPb = 0 TPa = 1 x TNa = y + FPb TPb = 1 y TNb = y FPb FNa = 0 FPa = x y FPb FNb = 0 TPa = 1 x FNb TNa = x FPb TPb = 1 y FNb TNb = y FPb FNa = FNb FPa = FPb x = y Maximal Combinations of Fairness Definitions B.6 Demographic Parity, Predictive Equality and Predictive Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb FPa = x y FPb TPa = TPb FPa = FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPb = y x y FPb + TPb FPb = x y FPb TPa = TPb FPa = FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y 0 = y x y FPb 1 = x TPa = TPb FPa = FPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y x = y FPb = 0 x = y FPb = 0 TPa = TPb FPa = FPb FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb x = y TPa = TPb FPa = FPb FNa = 1 x TPb TNa = x FNb = 1 y TPb TNb = y FPb = 0 TPa = TPb FPa = 0 B.7 Demographic Parity, Predictive Equality and False Omission Rate Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb TNa = x y TNb TNa = TNb FNa = FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb TNb = x y TNb TNa = TNb TPa = 1 + x + 1 y TPb Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb 1 = x TNa = TNb TPa = x + y + TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y x + y + TPb = y x y FPb + TPb x = y TNb = 0 TNa = TNb TPa = x + y + TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y x + y = y x y FPb x = y TNb = 0 TNa = TNb TPa = x + y + TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y 1 = 1 y FPb x = y x = y TNb = 0 TNa = TNb TPa = x + y + TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y y = FPb x = y x = y TNb = 0 TNa = TNb TPa = x + y + TPb FNa = 1 x TPb FPa = x TNb FNb = 1 y TPb FPb = y TNb x = y x = y TNa = TNb TPa = TPb FNa = 1 y TPb FPa = x FNb = 1 y TPb 0 = 0 FPb = y TNb = 0 TNa = 0 TPa = x + y + TPb Maximal Combinations of Fairness Definitions B.8 Demographic Parity, Predictive Equality and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb FPa = x y FPb TPa = TPb + y x 2 FPa = FPb + x y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb FPa = x y FPb + TPb = TPb + y x 2 x y FPb = FPb + x y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb FPa = x y FPb = y x y FPb = x y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb FPa = x y FPb 1 y FPb = 1 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb FPa = x y FPb FPb = y FNa = 1 x y x 2 TPb TNa = x 2 FNb = 1 y TPb TNb = y 2 TPa = y x 2 + TPb FPa = x FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb x = y Defrance & De Bie B.9 Demographic Parity, Predictive Equality and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = y x y FPb + TPb FPa = x y FPb FNa = FNb ( x y FPb FNb + 1) FPa = FPb ( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = x y FNb + y x y FPb FPa = x y FPb x y FPb = FPb ( x y FPb FNb + 1) FNa = FNb ( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb x y = ( x y FPb FNb + 1) FPb = 0 y x + FNb y x y FPb = FNb ( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y = x y FPb FNb FPb = 0 y FPb = FNb ( x y FPb FNb ) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb FPb FNb = y FPb = 0 x y = 0 1 1 y FPb = FNb 1 FPb FNb x = y Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb FPb FNb = y FPb = 0 x y = 0 y FPb = y FNb 1 FPb FNb x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb FPb FNb = y FPb = 0 x y = 0 y FPb y FNb FP 2 b + FNb FPb = y FNb x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb FPb FNb = y FPb = 0 x y = 0 y FPb FP 2 b + FNb FPb = 0 x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb FPb FNb = y FPb = 0 x = y y FPb + FNb = 0 x = y FPb = 0 Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb FPb FNb = y FPb = 0 x = y FPb FNb = y x = y FPb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb FNb = FPb y FPb = 0 x = y TPa = 1 x FNa TNa = x FPa TPb = 1 y FNb TNb = y FPb FNb = FPb y FNa = x(1 1 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb y x y FPb FPa = x y FPb FPa = 0 FPa = FPb TPa = 1 x TNa = 0 TPb = 1 y TNb = 0 FNb = 0 FNa = 0, FPb = y FPa = x TPa = 1 y FNb TNa = x TPb = 1 y FNb TNb = y FPb = 0 FNa = y x + FNb FPa = 0 TPa = 1 x FNb TNa = x FPb TPb = 1 y FNb TNb = y FPb x = y FNa = FNb FPa = FPb Maximal Combinations of Fairness Definitions B.10 Demographic Parity, Predictive Parity and False Omission Rate Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb TNa = TNb FNa = FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb x FPa = y FPb 1 x TPa = 1 y TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb x FPa = y FPa x TPa = y TPa FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb x = y x = y B.11 Demographic Parity, Predictive Parity and Overall Accuracy equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb FPa = FPb + x y 2 TPa = TPb + y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb FPa = FPa + x y 2 TPa = TPa + y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb 0 = x y FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb x = y x = y Defrance & De Bie B.12 Demographic Parity, Predictive Parity and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa = FPb FNa = FNb ( x y FPb FNb + 1) FPa = FPb ( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y 1 x FNa = 1 y FNb FPa = FPb FNa = FNb ( x y FPb FNb + 1) FPb = FPb ( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb FPa = FPb y x + FNb = FNb ( x y FPb FNb + 1) 1 = x y FPb FNb + 1 FPb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = y x + FNb FPa = FPb y x = FNb ( x y FPb FNb ) x = y FPb = 0 TPa = 1 x FNb TNa = x FPb TPb = 1 y FNb TNb = y FPb FNa = FNb FPa = FPb 0 = 0 x = y TPa = 1 y FNb TNa = x TPb = 1 y FNb TNb = y FNa = y x + FNb FPa = FPb = 0 y x = (x y) FPb = 0 Maximal Combinations of Fairness Definitions B.13 Demographic Parity, False Omission Rate Parity and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = TNb FNa = FNb FPa = FPb + x y 2 TPa = TPb + y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y x FPa = y FPb 1 x TPa = 1 y TPb FPa = FPb + x y 2 TPa = TPb + y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb TPa = y x + TPb x y + FPb = FPb + x y 2 y x + TPb = TPb + y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb TPa = y x + TPb x y = x y 2 y x = y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb TPa = y x + TPb x = y x = y FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb FPa = FPb TPa = TPb x = y B.14 Demographic Parity, False Omission Rate Parity and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb FNa = FNb FNa = FNb( x y FPb FNb + 1) FPa = FPb( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb FNa = FNb FNb = FNb( x y FPb FNb + 1) x y + FPb = FPb( x y FPb FNb + 1) Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb FNa = FNb 1 = x y FPb FNb + 1 FNb = 0 x y = FPb x y FPb FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb FNa = FNb 0 = x y FPb FNb FNb = 0 1 = FPb FPb FNb x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb FNa = FNb x = y FNb = 0 FPb FNb = FPb x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y + FPb FNa = FNb x = y FNb = 0 FNb = 0 x = y TPa = 1 x FNb TNa = x FPb TPb = 1 y FNb TNb = y FPb FPa = FPb FNa = FNb x = y TPa = 1 x TNa = y FPb TPb = 1 y TNb = y FPb FPa = x y + FPb FNa = 0 FNb = 0 B.15 Demographic Parity, Overall accuracy Equality and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb + x y 2 TPa = TPb + y x 2 FNa = FNb( x y FPb FNb + 1) FPa = FPb( x y FPb FNb + 1) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb + x y 2 FNa = y x + FNb y x 2 y x + FNb y x 2 = FNb( x y FPb FNb + 1) 2 = FPb( x y FPb FNb + 1) Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb + x y 2 FNa = y x + FNb y x 2 = FNb x y FPb FNb x y 2 = FPb x y FPb FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb + x y 2 FNa = y x + FNb y x 2 = FNb 1 FPb FNb x = y 1 2 = FPb 1 FPb FNb x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb + x y 2 FNa = y x + FNb y x 2 = FNb x = y FPb FNb 2 = FPb x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb + x y 2 FNa = FNb + y x 2 x = y FNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb + x y 2 FNa = FNb + y x 2 FPb = FNb = 0 x = y FNb = FPb = 0 x = y TPa = 1 x y x 2 TNa = x x y 2 TPb = 1 y TNb = y FPa = x y 2 FNa = y x 2 FPb = FNb = 0 TPa = 1 x FNa TNa = x FPa TPb = 1 y FNb TNb = y FPb FPa = FPb FNa = FNb x = y Defrance & De Bie B.16 Equalised odds and Predictive Parity/Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y FPb TPa = 1 x 1 y TPb FPa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb 1 x 1 y FPb = x y FPb FPa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb 1 x 1 y = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb y xy = x xy FPb = 0 FPa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb y = x FPb = 0 FPa = 1 x FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb y = x FPa = FPb FNa = 1 x 1 x 1 y TPb TNa = x FNb = 1 y TPb TNb = y TPa = 1 x 1 y TPb FPb = 0 FPa = 0 Maximal Combinations of Fairness Definitions B.17 Equalised odds and False Omission Rate Parity TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb TNa = x y TNb FNa = 1 x 1 y FNb TNa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb TNa = x y TNb x y TNb = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb TNa = x y TNb x y = 1 x 1 y TNb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb TNa = x y TNb x = y TNb = 0 TPa = 1 x FNb FPa = x TNb TPb = 1 y FNb FPb = y TNb FNa = FNb TNa = TNb x = y TPa = 1 x 1 x 1 y FNb FPa = x TPb = 1 y FNb FPb = y FNa = 1 x 1 y FNb TNa = 0 TNb = 0 B.18 Equalised odds and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y FPb TPa = 1 x 1 y TPb y x 1 y TPb = TNb TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y FPb 1 x 1 y TPb = 1 x 1 y TPb y x 1 y TPb = y x Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = x y FPb TPb = 1 y y TNb x = y FNa = 1 x 1 x y TNb TNa = x x y FPb FNb = 1 y 1 y y TNb TNb = y FPb TPa = 1 x y TNb FPa = x y FPb TPb = 1 y FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb FPa = FPb x = y B.19 Equal Opportunity, False Omission Rate Parity and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb TNa = 1 x 1 y TNb FNa = 1 x 1 y FNb TNa = TNb y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb TNa = 1 x 1 y TNb 1 x 1 y TNb = TNb y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb TNa = 1 x 1 y TNb 1 x 1+y 1 y TNb = y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FNa = 1 x 1 y FNb TNa = 1 x 1 y TNb y x 1 y TNb = y x Maximal Combinations of Fairness Definitions TPa = 1 x FNa FPa = x TNa TPb = 1 y FNb FPb = y TNb FNa = 1 x 1 y FNb TNa = 1 x 1 y TNb TNb = TPb x = y TPa = 1 x 1 x 1 y FNb FPa = x 1 x 1 y TNb TPb = 1 y FNb FPb = y TNb FNa = 1 x 1 y FNb TNa = 1 x 1 y TNb TNb = TPb = 0 x = y B.20 Equal Opportunity, False Omission Rate Parity and Predictive Parity/Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = 1 x 1 y FPb TPa = 1 x 1 y TPb TNa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb x TNa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x 1 y(y TNb) = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x FNa = 1 x TPb FPa = x TNb FNb = 1 y TPb FPb = y TNb TPa = TPb TNa = TNb x = y Defrance & De Bie B.21 Equal Opportunity, Overall accuracy Equality and Predictive Parity/Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb FPa = 1 x 1 y FPb TPa = 1 x 1 y TPb TNa = TNb y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb x TNa = 1 x TNa = TNb y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x 1 y TNb = TNb y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x 1 y TNb = y x 1 y TPb x + 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x 1 y(y TNb) y x 1 y TNb = y x 1 y TPb x + 1 x Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x TNb = TPb 1 y 1 y 1 y y xy x = y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x TNb = TPb 1 y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x 1 y TPb TNa = x 1 x TNb = TPb + 1 y x( x + y) x = y FNa = 1 x TPa TNa = x FPa FNb = 1 y TPb TNb = y FPb TPa = 1 x 1 y TPb TNa = 1 1 x 1 y TPb TNb = 1 TPb FNa = 1 x TPa TNa = x FPa FNb = 1 y TPb TNb = y FPb TPa = TPb TNa = TNb x = y FNa = 0 FPa = 0 FNb = 0 TNb = 0 TPa = 1 x TNa = x TNb = y TPb = 1 y FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb TPa = TPb TNa = TNb x = y Defrance & De Bie B.22 Predictive Equality, Predictive Parity and False Omission Rate Parity/Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb TPa = x y TPb FPa = x y FPb FNa = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb FNa = x y FNb 1 x FNa = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb FNa = x y FNb 1 x x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = x y FPb FNa = x y FNb 1 = x TPa = 1 x FNb TNa = x FPb TPb = 1 y FNb TNb = y FPb y = x FPa = FPb FNa = FNb B.23 Predictive Equality, Predictive Parity and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa = x y TPb TNa = x x y(TPb TPa) TNb = y x y(TPb TPa) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa = x y TPb TNa = x x y(TPb x TNb = y x y(TPb x Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa = x y TPb TNa = x x y(y x TNb = y x y(y x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa = x y TPb TNa = x y TPb TNb = TPb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = x y TPb TNa = x y TPb TNb = TPb x FNa = 1 x FPa = x FNb = 1 y FPb = y TPa = x TNb = TPb = 0 1 = 1 B.24 Predictive Equality, Overall accuracy Equality and False Omission Rate Parity/Treatment equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb FNa = x y FNb TNa = x x y(TPb TPa) TNb = y x y(TPb TPa) TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa = 1 x y TPb TNa = x x y(TPb 1 + x TNb = y x y(TPb 1 + x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa = 1 x y TPb TNa = x x y(x y TNb = y x y(x y TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y TNb TPa = 1 x y TPb TNa = x y TPb TNb = 1 TPb Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = 1 x y TPb TNa = x y TPb TNb = 1 TPb x y x FNa = 1 x TPa FPa = x TNa FNb = 1 y TPb FPb = y TNb TPa = 1 x y TPb TNa = x y TPb TNb = 1 TPb 1 = 1 FNa = 0 FPa = 0 FNb = 0 FPb = 0 TPa = 1 x TNa = x TNb = y TPb = 1 y 1 = 1 B.25 Predictive Parity, False Omission Rate Parity and Overall accuracy Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb FNa = FNb TNa TNb TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa (y TNb) = TPb (x TNa) (1 x TPa) = (1 y TPb) TNa TNb TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb x TNa y TNb (1 x TPb x TNa y TNb ) TNb = (1 y TPb) TNa TPa + TNa = TPb + TNb Maximal Combinations of Fairness Definitions TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb x TNa y TNb TPb TNb x TNa y TNb + TPb TNa = TNb + x TNb + TNa y TNa TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb x TNa y TNb TPb(TNa TNb x TNa y TNb ) = (x 1)TNb + (1 y)TNa TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb x TNa y TNb TPb(y TNa TNa TNb x TNb+TNb TNa y TNb ) = (x 1)TNb + (1 y)TNa TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb x TNa y TNb TPb(y TNa x TNb y TNb ) = (x 1)TNb + (1 y)TNa TPa + TNa = TPb + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = (x 1)(x TNa) y TNa x TNb TNb + (1 y)(x TNa) y TNa x TNb TNa TPb = (x 1)(y TNb) y TNa x TNb TNb + (1 y)(y TNb) y TNa x TNb TNa TPa + TNa = TPb + TNb Defrance & De Bie TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = (x 1)(x TNa) y TNa x TNb TNb + (1 y)(x TNa) y TNa x TNb TNa TPb = (x 1)(y TNb) y TNa x TNb TNb + (1 y)(y TNb) y TNa x TNb TNa (x 1)(x TNa) y TNa x TNb TNb + (1 y)(x TNa) y TNa x TNb TNa + TNa = (x 1)(y TNb) y TNa x TNb TNb + (1 y)(y TNb) y TNa x TNb TNa + TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = (1 x)(x TNa) y TNa x TNb TNb + (1 y)(x TNa) y TNa x TNb TNa TPb = (1 x)(y TNb) y TNa x TNb TNb + (1 y)(y TNb) y TNa x TNb TNa (x2 x TNa x + TNa)TNb + (x xy TNa + y TNa)TNa+ (y TNa x TNb)TNa = (xy x TNb y+ < tn b)TNb+ (y TNb y2 + y TNb)TNa + (y TNa x TNb)TNb TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = (1 x)(x TNa) y TNa x TNb TNb + (1 y)(x TNa) y TNa x TNb TNa TPb = (1 x)(y TNb) y TNa x TNb TNb + (1 y)(y TNb) y TNa x TNb TNa (xy 2x TNb y + TNb + y TNa x2 + x TNa + x TNa)TNb = (x xy TNa + 2y TNa x TNb y + TNb + y2 y TNb)TNa TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = (1 x)(x TNa) y TNa x TNb TNb + (1 y)(x TNa) y TNa x TNb TNa TPb = (1 x)(y TNb) y TNa x TNb TNb + (1 y)(y TNb) y TNa x TNb TNa ( 2x + 1)TN2 b + (xy y + x + 2y TNa x2 + 2x TNa 2TNa)TNb +(xy x + TNa 2y TNa + y y2)TNa = 0 Maximal Combinations of Fairness Definitions B.26 Predictive Parity, False Omission Rate Parity and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TPa = TPb FPa FPb FNa = FNb TNa TNb FNa = 1 x 1 y FNb FPa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y 1 x 1 y TPb = TPb FPa FPb 1 x 1 y FNb = FNb TNa TNb FNa = 1 x 1 y FNb FPa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y 1 x 1 y FPb = FPa TPb = 0 1 y TNb FNb = 0 1 y FNb FPa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = 1 x 1 y TNb FNb = 0 1 y FNb TNa = x 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x 1 x 1 y TNb TNa = 1 x 1 y TNb FNa = 1 x TPa = 1 x FPa = x TNa TPb = 1 y FPb = y TNb TNa = x 1 x 1 y TNb FNb = 0 FNa = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y 1 x 1 y TNb = x 1 x 1 y TNb TNa = 1 x 1 y TNb FNa = 1 x TPa = 1 x FPa = x TNa TPb = 1 y FPb = y TNb TNa = x 1 x 1 y TNb FNb = 0 FNa = 0 Defrance & De Bie TPa = 1 x FNb FPa = x TNb TPb = 1 y FNb FPb = y TNb x = y TNa = TNb FNa = FNb TPa = 1 x FPa = 1 x 1 y TNb TPb = 1 y FPb = y TNb TNa = x 1 x 1 y TNb FNb = 0 FNa = 0 B.27 Predictive Parity, Overall accuracy Equality and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb FNa = FNb TPa = 1 x 1 y TPb FPa = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb 1 x TPa = 1 y TPb TPa = 1 x 1 y TPb FPb = 1 x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb TPa = y x + TPb y x TPb = 1 x 1 y TPb x = y FPb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb TPa = y x + TPb y x = y x 1 y TPb x = y FPb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb TPa = y x + TPb TPb = 1 y x = y x = y FPb = 0 FNa = 0 TNa = x FNb = 0 TNb = y FPa = 0 TPa = 1 x TPb = 1 y FPb = 0 FNa = 1 x TPb TNa = x FPb FNb = 1 y TPb TNb = y FPb FPa = FPb TPa = TPb x = y Maximal Combinations of Fairness Definitions B.28 False Omission Rate Parity, Overall accuracy Equality and Treatment Equality TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y FPa = FPb FNa = FNb TNa = x y TNb FNa = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y x TNa = y TNb FNa = FNb TNa = x y TNb FNb = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y + TNb FNa = FNb x y + TNb = x y TNb 1 = x TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y + TNb FNa = FNb x y = x y y TNb x = y FNb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y + TNb FNa = FNb 1 = 1 y TNb x = y x = y FNb = 0 TPa + FNa = 1 x TNa + FPa = x TPb + FNb = 1 y TNb + FPb = y TNa = x y + TNb FNa = FNb TNb = y x = y x = y FNb = 0 TPa = 1 x FPa = 0 TPb = 1 y FPb = 0 TNa = x FNa = 0 TNb = y FNb = 0 TPa = 1 x FNb FPa = x TNb TPb = 1 y FNb FPb = y TNb TNa = TNb FNa = FNb x = y Defrance & De Bie Appendix C. p%-rule Feasibility C.1 Equalised Odds 1 x 1 y TPb + x y FPb p TPb + p FPb TPb + FPb p1 x 1 y TPb + px Case 1: x = y TPb FPb This is always true, thus for x = y equalised odds satisfies the p%-rule for all p. Case 2: p < 1 x 1 y and p < 1 y y yp x 1 x p + yp FPb y xp y 1 y p + xp FPb Case 3: p > 1 x 1 y From this condition follows that x > y as p 1 and x = y. Furthermore this means that p < 1 y y yp x 1 x p + yp FPb y xp y 1 y p + xp FPb Case 4: p > 1 y 1 x Similar to case 3, we can derive that y > x and p < 1 x y yp x 1 x p + yp FPb y xp y 1 y p + xp FPb Case 5: p = 1 x 1 y From this condition we can derive that x y and thus p < 1 y y xp y 1 y p + xp FPb Case 6: p = 1 y 1 x From this we can derive that x y and thus p < 1 x y yp x 1 x p + yp FPb Maximal Combinations of Fairness Definitions C.2 Equal Opportunity and Predictive Parity 1 x 1 y TPb + 1 x 1 y FPb p TPb + p FPb TPb + FPb p1 x 1 y TPb + p1 x Case 1: x = y TPb FPb This is always true, thus for x = y all p values are satisfied. Case 2: p 1 x 1 y and p 1 y 1 x TPb FPb This is always for all p values that fall within these constraints. Case 3: p > 1 x 1 y or p > 1 y 1 x TPb FPb This can only be satisfied if both variables are 0, which we do not considered as an acceptable constraint, thus these p-values cannot be satisfied. C.3 Equal Opportunity and False Omission Rate Parity 1 p + p FNb + p TNb 1 x 1 y FNb + 1 x 1 y p FNb + 1 x 1 y p TNb FNb + TNb Case 1: x = y 1 FNb TNb This condition is always satisfied, thus for equal base rates this combination of fairness constraints will also satisfy demographic parity. Case 2: p < 1 x 1 y and p < 1 y (1 p)(1 y) p yp 1 + x + FNb TNb (1 p)(1 y) p xp 1 + y + FNb TNb Case 3: p > 1 x 1 y and thus p < 1 y (1 p)(1 y) p yp 1 + x + FNb TNb (1 p)(1 y) p xp 1 + y + FNb TNb If p = 1 than the fraction in the second equation becomes zero. In order to satisfy that equation under the condition p = 1 then it requires FNb = TNb = 0, which is not Defrance & De Bie acceptable. Thus the equation is not possible but for larger p, the ranges for FNb and TNb become undesirable. Case 4: p > 1 y 1 x and thus p < 1 x (1 p)(1 y) p yp 1 + x + FNb TNb (1 p)(1 y) p xp 1 + y + FNb TNb If p = 1 than the fraction in the first equation becomes zero. In order to satisfy that equation under the condition p = 1 then it requires FNb = TNb = 0, which is not acceptable. Thus the equation is not possible but for larger p, the ranges for FNb and TNb become undesirable. Case 5: p = 1 x 1 y (1 p)(1 y) p xp 1 + y + FNb TNb Case 6: p = 1 y 1 x (1 p)(1 y) p yp 1 + x + FNb TNb C.4 Equalised Opportunity and Overall accuracy Equality x py + TNb(1 x p + 2py xp y x ) TNa(1 2x p + yp + y (y xp) + TNa(1 y p + 2xp yp y x ) TNb(1 2y p + xp + x Case 1: x = y x + TPb TNb This statement is always true, thus in this case the p%-rule will always be satisfied for every p. Case 2: y > x Case 2a: p 1 2x+y 1 y p 1 2y+x (x py)(y x) 1 2x p + yp + y + 1 x p + 2py xp 1 2x p + yp + y TNb TNa (y xp)(y x) 1 2y p + xp + x + 1 y p + 2xp yp 1 2y p + xp + x TNa TNb Case 2b: p 1 2x+y 1 y p > 1 2y+x (x py)(y x) 1 2x p + yp + y + 1 x p + 2py xp 1 2x p + yp + y TNb TNa (y xp)(y x) 1 2y p + xp + x + 1 y p + 2xp yp 1 2y p + xp + x TNa TNb Maximal Combinations of Fairness Definitions In the case that p = 1 then these equations would simplify to TNa = TNb + x y 2 . While this equation does not violate this relationship will make TPa = 1 x 2 and TPb = 1 y 2 , which is a set value for these variables, making that for p = 1 this combination is not possible. Case 3: y < x Case 3a: p 1 2x+y 1 y p 1 2y+x (x py)(y x) 1 2x p + yp + y + 1 x p + 2py xp 1 2x p + yp + y TNb TNa (y xp)(y x) 1 2y p + xp + x + 1 y p + 2xp yp 1 2y p + xp + x TNa TNb Case 3b: p > 1 2x+y 1 y p 1 2y+x (x py)(y x) 1 2x p + yp + y + 1 x p + 2py xp 1 2x p + yp + y TNb TNa (y xp)(y x) 1 2y p + xp + x + 1 y p + 2xp yp 1 2y p + xp + x TNa TNb In the case that p = 1 then these equations would simplify to TNa = TNb + x y 2 . While this equation does not violate this relationship will make TPa = 1 x 2 and TPb = 1 y 2 , which is a set value for these variables, making that for p = 1 this combination is not possible. C.5 Equal Opportunity and Treatment Equality 1 y FNb + 1 x 1 y FPb p(1 y FNb) + p FPb 1 y FNb + FPb p(1 x 1 x 1 y FNb) + p1 x Case 1: p > 1 x 1 y p > 1 y 1 x (1 y) + FNb FPb This is impossible to satisfy, so it is not possible to satisfy for p > 1 x Case 2: p 1 x 1 x (1 y) + FNb FPb This always holds true, so for p 1 x 1 x this sets of constraints always satisfies the p%-rule. C.6 Predictive Equality and Predictive Parity x y TPb + x y p TPb + p FPb TPb + FPb px Case 1: p > x Defrance & De Bie This is impossible to satisfy, thus p > x x is not possible. Case 2: p x This always hold true, thus this combinations for constraints always satisfies for p x C.7 Predictive Equality and False Omission Rate Parity 1 p + p FNb + p TNb x y TNb FNb + TNb Case 1: p x py x + FNb TNB y yp y xp FNb TNb The first equation is always simplified so this simplifies to: y yp y xp FNb TNb This equation can only be satisfied for p = 1 if FNb = 0 and TNb = 0. Case 2: p < x py x + FNb TNB y yp y xp FNb TNb This first equation cannot be satisfied for p = 1, unless FNb = 0 and TNb = 0. Case 3: p < x py x + FNb TNB y yp y xp FNb TNb C.8 Predictive Equality and Treatment Equality y FPb p(1 y FNb) + p FPb 1 y FNb + FPb p(1 x x y FNb) + px Case 1: p x x py) + FNb FPb Maximal Combinations of Fairness Definitions y px) + FNb FPb The first equation is always satisfied. The second equation cannot be satisfied for p = 1 unless FNb = 0 and FPb = y Case 2: p y x py) + FNb FPb y px) + FNb FPb The second equation is always satisfied. The first equation cannot be satisfied for p = 1 unless FNb = 0 and FPb = y Case 3: p < y x py) + FNb FPb y px) + FNb FPb C.9 Predictive Parity and Overall accuracy Equality (FPa p FPb)( y x FPa FPb ) 2(FPa p FPb) (FPb p FPa)( y x FPa FPb ) 2(FPb p FPa) Case 1: FPa p FPb FPa FPb FPb p FPa 2 + FPa FPb If y x then this statement is always true. For p = 1 the constraints would require that FPa = FPb. Case 2: FPa p FPb FPa FPb FPb p FPa 2 + FPa FPb 2 + FPa FPb This simplifies to: y x 2 + FPa = FPb Case 3: FPa p FPb FPa FPb FPb p FPa 2 + FPa FPb Case 4: FPa p FPb FPa FPb FPb p FPa 2 + FPa FPb Defrance & De Bie 2 + FPa FPb This simplifies to: y x 2 + FPa = FPb C.10 Predictive Parity and Treatment Equality 1 x p + px + (p 1 x 1 y )FNb (p 1 x 1 y p + xp + (p1 x 1 y 1)FNb (p1 x Case 1: p > 1 x (1 y) + FNb FPb (1 y) + FNb FPb The first equation cannot be satisfied, it is thus impossible to satisfy the p%-rule in these conditions. Case 2: p 1 x 1 y p > 1 y (1 y) + FNb FPb (1 y) + FNb FPb The second equation cannot be satisfied, it is thus impossible to satisfy the p%-rule in these conditions. Case 3: p 1 x (1 y) + FNb FPb This equation is always satisfied for all values for p. C.11 False Omission Rate Parity and Overall Accuracy Equality 1 1 FNb TNb ( FNb + TNb 1 FNb TNb )(1 + x + y TNb FNb ) p 1 FNb TNb p(1 (FNb + TNb)(1 + x y TNb FNb ) In the case that p = 1 this equation will simplify to: These equations can only be satisfied for TNb = 0 and FNb = 0, however this is not possible due to the original formulation of the equations. Maximal Combinations of Fairness Definitions C.12 False Omission Rate Parity and Treatment Equality y )FNb (p x y 1)FNb (px Case 1: p x x yp + FNb TNb y xp + FNb TNb This first equation is always satisfied as p > x y, the second equation however cannot be satisfied for p = 1, unless FNb = 0 and TNb = 0. Case 2: p < x x yp + FNb TNb y xp + FNb TNb This second equation is always satisfied as p > y x, the first equation however cannot be satisfied for p = 1, unless FNb = 0 and TNb = 0. Case 3: p < x x yp + FNb TNb y xp + FNb TNb C.13 Overall accuracy Equality and Treatment Equality 1 x p + yp 1 p + FPb FNb 1 p + FPb FNb These equations hold for all cases. In the special case that p goes to 1, then the first equation requires x y for the inequality to hold, however the second equation would require y x. Thus these equations can only hold for p = 1 if x = y. Agarwal, A., Lohia, P., Nagar, S., Dey, K., & Saha, D. (2018). Automated test generation to detect individual discrimination in ai models. Agarwal, S. (2021). Trade-offs between fairness and privacy in machine learning. In IJCAI 2021 Workshop on AI for Social Good. Defrance & De Bie Angwin, J., Larson, J., Mattu, S., & Kirchner, L. (2016). Machine bias. Pro Publica. Australian Law Reform Commission (2016). Procedural fairness: the duty and its content. Barocas, S., Hardt, M., & Narayanan, A. (2019). Fairness and Machine Learning: Limitations and Opportunities. fairmlbook.org. Beigang, F. (2023). Yet Another Impossibility Theorem in Algorithmic Fairness. Minds and Machines. Bell, A., Bynum, L., Drushchak, N., Zakharchenko, T., Rosenblatt, L., & Stoyanovich, J. (2023). The possibility of fairness: Revisiting the impossibility theorem in practice. In Proceedings of the 2023 ACM Conference on Fairness, Accountability, and Transparency, FAcc T 23, page 400 422, New York, NY, USA. Association for Computing Machinery. Berk, R., Heidari, H., Jabbari, S., Kearns, M., & Roth, A. (2021). Fairness in criminal justice risk assessments: The state of the art. Sociological Methods & Research, 50(1):3 44. Black, E., Yeom, S., & Fredrikson, M. (2020). Fliptest: Fairness testing via optimal transport. In Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency, FAT* 20, page 111 121, New York, NY, USA. Association for Computing Machinery. Buyl, M., Defrance, M., & Bie, T. D. (2024). fairret: a framework for differentiable fairness regularization terms. (ar Xiv:2310.17256). ar Xiv:2310.17256 [cs]. Caton, S. & Haas, C. (2024). Fairness in machine learning: A survey. ACM Computing Surveys, 56(7):1 38. Chouldechova, A. (2016). Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. Corbett-Davies, S., Pierson, E., Feller, A., Goel, S., & Huq, A. (2017). Algorithmic decision making and the cost of fairness. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 17, page 797 806, New York, NY, USA. Association for Computing Machinery. Cummings, R., Gupta, V., Kimpara, D., & Morgenstern, J. (2019). On the compatibility of privacy and fairness. In Adjunct Publication of the 27th Conference on User Modeling, Adaptation and Personalization, UMAP 19 Adjunct, page 309 315, New York, NY, USA. Association for Computing Machinery. Dwork, C., Hardt, M., Pitassi, T., Reingold, O., & Zemel, R. (2012). Fairness through awareness. ITCS 12, page 214 226, New York, NY, USA. Association for Computing Machinery. Equal employment opportunity commission (1978). Section 1607.4, uniform guidelines on employee selection procedure. Maximal Combinations of Fairness Definitions Fabris, A., Messina, S., Silvello, G., & Susto, G. A. (2022). Algorithmic fairness datasets: the story so far. Data Mining and Knowledge Discovery, 36(6):2074 2152. Galhotra, S., Brun, Y., & Meliou, A. (2017). Fairness testing: Testing software for discrimination. In Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering, ESEC/FSE 2017, page 498 510, New York, NY, USA. Association for Computing Machinery. Ghosh, A., Genuit, L., & Reagan, M. (2021). Characterizing intersectional group fairness with worst-case comparisons. In Lamba, D. & Hsu, W. H., editors, Proceedings of 2nd Workshop on Diversity in Artificial Intelligence (AIDBEI), volume 142 of Proceedings of Machine Learning Research, pages 22 34. PMLR. Goethals, S., Martens, D., & Calders, T. (2023). Precof: counterfactual explanations for fairness. Machine Learning. Grari, V., Lamprier, S., & Detyniecki, M. (2023). Adversarial learning for counterfactual fairness. Machine Learning, 112(3):741 763. Grgic-Hlaca, N., Redmiles, E. M., Gummadi, K. P., & Weller, A. (2018). Human perceptions of fairness in algorithmic decision making: A case study of criminal risk prediction. In Proceedings of the 2018 World Wide Web Conference, WWW 18, page 903 912, Republic and Canton of Geneva, CHE. International World Wide Web Conferences Steering Committee. Hardt, M., Price, E., & Srebro, N. (2016). Equality of opportunity in supervised learning. In Lee, D., Sugiyama, M., Luxburg, U., Guyon, I., & Garnett, R., editors, Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc. Hsu, B., Mazumder, R., Nandy, P., & Basu, K. (2022). Pushing the limits of fairness impossibility: Who's the fairest of them all? In Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., & Oh, A., editors, Advances in Neural Information Processing Systems, volume 35, pages 32749 32761. Curran Associates, Inc. Kamishima, T., Akaho, S., & Sakuma, J. (2011). Fairness-aware learning through regularization approach. In 2011 IEEE 11th International Conference on Data Mining Workshops, page 643 650. Kleinberg, J. M., Mullainathan, S., & Raghavan, M. (2016). Inherent trade-offs in the fair determination of risk scores. Co RR, abs/1609.05807. Kusner, M. J., Loftus, J., Russell, C., & Silva, R. (2017). Counterfactual fairness. In Guyon, I., Luxburg, U. V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., & Garnett, R., editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc. Lamont, J. & Favor, C. (2017). Distributive Justice. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter 2017 edition. Defrance & De Bie Liu, S. & Vicente, L. N. (2022). Accuracy and fairness trade-offs in machine learning: a stochastic multi-objective approach. Computational Management Science, 19(3):513 537. Luo, L., Nakao, Y., Chollet, M., Inakoshi, H., & Stumpf, S. (2024). Earn fairness: Explaining, asking, reviewing, and negotiating artificial intelligence fairness metrics among stakeholders. (ar Xiv:2407.11442). ar Xiv:2407.11442 [cs]. Mehrabi, N., Morstatter, F., Saxena, N., Lerman, K., & Galstyan, A. (2021). A survey on bias and fairness in machine learning. ACM Comput. Surv., 54(6). Mitchell, S., Potash, E., Barocas, S., D Amour, A., & Lum, K. (2021). Algorithmic fairness: Choices, assumptions, and definitions. Annual Review of Statistics and Its Application, 8(1):141 163. Padala, M. & Gujar, S. (2020). Fnnc: Achieving fairness through neural networks. In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, page 2277 2283, Yokohama, Japan. International Joint Conferences on Artificial Intelligence Organization. Park, H., Ahn, D., Hosanagar, K., & Lee, J. (2022). Designing fair ai in human resource management: Understanding tensions surrounding algorithmic evaluation and envisioning stakeholder-centered solutions. In CHI Conference on Human Factors in Computing Systems, page 1 22, New Orleans LA USA. ACM. Pessach, D. & Shmueli, E. (2022). A review on fairness in machine learning. ACM Comput. Surv., 55(3). Pinz on, C., Palamidessi, C., Piantanida, P., & Valencia, F. (2023). On the incompatibility of accuracy and equal opportunity. Machine Learning. Pleiss, G., Raghavan, M., Wu, F., Kleinberg, J., & Weinberger, K. Q. (2017). On fairness and calibration. In Guyon, I., Luxburg, U. V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., & Garnett, R., editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc. Rahman, M. M., Pan, S., & Foulds, J. R. (2024a). Towards a unifying human-centered ai fairness framework. In Proceedings of the 2024 International Conference on Information Technology for Social Good, Good IT 24, page 88 92, New York, NY, USA. Association for Computing Machinery. Rahman, M. M., Pan, S., & Foulds, J. R. (2024b). Towards a unifying human-centered ai fairness framework. In Proceedings of the 2024 International Conference on Information Technology for Social Good, Good IT 24, page 88 92, New York, NY, USA. Association for Computing Machinery. Rosenblatt, L. & Witter, R. T. (2023). Counterfactual fairness is basically demographic parity. In Proceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence, AAAI 23/IAAI 23/EAAI 23. AAAI Press. Maximal Combinations of Fairness Definitions Roth, P. L., Bobko, P., & Switzer, F. S. r. (2021). Modeling the behavior of the 4/5ths rule for determining adverse impact: reasons for caution. J Appl Psychol, 91(3):507 522. Ruf, B. & Detyniecki, M. (2021). Towards the right kind of fairness in AI. Co RR, abs/2102.08453. Saleiro, P., Kuester, B., Hinkson, L., London, J., Stevens, A., Anisfeld, A., Rodolfa, K. T., & Ghani, R. (2019). Aequitas: A bias and fairness audit toolkit. Sharma, S., Henderson, J., & Ghosh, J. (2020). Certifai: A common framework to provide explanations and analyse the fairness and robustness of black-box models. In Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, AIES 20. ACM. Verma, S. & Rubin, J. (2018). Fairness definitions explained. In 2018 IEEE/ACM International Workshop on Software Fairness (Fair Ware), pages 1 7. Zemel, R., Wu, Y., Swersky, K., Pitassi, T., & Dwork, C. (2013). Learning fair representations. In Proceedings of the 30th International Conference on Machine Learning, page 325 333. PMLR.